The document discusses arithmetic sequences and provides examples to illustrate how to determine if a sequence is arithmetic, derive the specific formula for an arithmetic sequence from the general formula, and use the specific formula to calculate future terms. It defines an arithmetic sequence as one where the terms follow a linear formula of an = d*n + c. Examples show how to identify the common difference d between terms and plug into the general formula along with the first term a1 to derive the specific formula for different sequences.

1. Arithmetic Sequences

2. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Arithmetic Sequences

3. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Arithmetic Sequences

4. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Fact: If a1, a2 , a3 , …is an arithmetic sequence and that
an = d*n + c then the difference between any two
neighboring terms is d, i.e. ak+1 – ak = d.
Arithmetic Sequences

5. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Fact: If a1, a2 , a3 , …is an arithmetic sequence and that
an = d*n + c then the difference between any two
neighboring terms is d, i.e. ak+1 – ak = d.
Arithmetic Sequences
In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

6. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Fact: If a1, a2 , a3 , …is an arithmetic sequence and that
an = d*n + c then the difference between any two
neighboring terms is d, i.e. ak+1 – ak = d.
Arithmetic Sequences
The following theorem gives the converse of the above fact
and the main formula for arithmetic sequences.
In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

7. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Fact: If a1, a2 , a3 , …is an arithmetic sequence and that
an = d*n + c then the difference between any two
neighboring terms is d, i.e. ak+1 – ak = d.
Arithmetic Sequences
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 – an = d for all n, then a1, a2, a3,… is an arithmetic
sequence
The following theorem gives the converse of the above fact
and the main formula for arithmetic sequences.
In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

8. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Fact: If a1, a2 , a3 , …is an arithmetic sequence and that
an = d*n + c then the difference between any two
neighboring terms is d, i.e. ak+1 – ak = d.
Arithmetic Sequences
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 – an = d for all n, then a1, a2, a3,… is an arithmetic
sequence and the formula for the sequence is
an = d(n – 1) + a1.
The following theorem gives the converse of the above fact
and the main formula for arithmetic sequences.
In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

9. A sequence a1, a2 , a3 , … is an arithmetic sequence
if an = d*n + c, i.e. it is defined by a linear formula.
Example A. The sequence of odd numbers
a1= 1, a2= 3, a3= 5, a4= 7, …
is an arithmetic sequence because an = 2n – 1.
Fact: If a1, a2 , a3 , …is an arithmetic sequence and that
an = d*n + c then the difference between any two
neighboring terms is d, i.e. ak+1 – ak = d.
Arithmetic Sequences
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 – an = d for all n, then a1, a2, a3,… is an arithmetic
sequence and the formula for the sequence is
an = d(n – 1) + a1.
This is the general formula of arithmetic sequences.
The following theorem gives the converse of the above fact
and the main formula for arithmetic sequences.
In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

10. Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

11. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

12. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

13. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

14. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

15. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
we get
an = 3(n – 1) + 2
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

16. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
we get
an = 3(n – 1) + 2
an = 3n – 3 + 2
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

17. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
we get
an = 3(n – 1) + 2
an = 3n – 3 + 2
an = 3n – 1 the specific formula.
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

18. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
we get
an = 3(n – 1) + 2
an = 3n – 3 + 2
an = 3n – 1 the specific formula.
c. Find a1000.
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

19. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
we get
an = 3(n – 1) + 2
an = 3n – 3 + 2
an = 3n – 1 the specific formula.
c. Find a1000.
Set n = 1000 in the specific formula,
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

20. Example B. Given the sequence 2, 5, 8, 11, …
a. Verify it is an arithmetic sequence.
It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.
b. Find the (specific) formula that represents this sequence.
Plug a1 = 2 and d = 3, into the general formula
an = d(n – 1) + a1
we get
an = 3(n – 1) + 2
an = 3n – 3 + 2
an = 3n – 1 the specific formula.
c. Find a1000.
Set n = 1000 in the specific formula, we get
a1000 = 3(1000) – 1 = 2999.
Arithmetic Sequences
Given the description of an arithmetic sequence, we use the
general formula to find the specific formula for that sequence.

21. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.

22. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

23. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1,
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

24. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

25. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula,
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

26. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

27. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

28. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

29. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

30. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
an = -4(n – 1) + 25
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

31. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
an = -4(n – 1) + 25
an = -4n + 4 + 25
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

32. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
an = -4(n – 1) + 25
an = -4n + 4 + 25
an = -4n + 29
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.

33. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
an = -4(n – 1) + 25
an = -4n + 4 + 25
an = -4n + 29
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.
To find a1000, set n = 1000 in the specific formula

34. Arithmetic Sequences
To use the arithmetic general formula to find the specific
formula, we need the first term a1 and the difference d.
Set d = –4 in the general formula an = d(n – 1) + a1, we get
an = –4(n – 1) + a1.
Set n = 6 in this formula, we get
a6 = -4(6 – 1) + a1 = 5
-20 + a1 = 5
a1 = 25
To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1
an = -4(n – 1) + 25
an = -4n + 4 + 25
an = -4n + 29
Example C. Given a1, a2 , a3 , …an arithmetic sequence with
d = -4 and a6 = 5, find a1, the specific formula and a1000.
To find a1000, set n = 1000 in the specific formula
a1000 = –4(1000) + 29 = –3971

35. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Arithmetic Sequences

36. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1,
Arithmetic Sequences

37. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
Arithmetic Sequences

38. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Arithmetic Sequences

39. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39

40. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

41. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

42. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

43. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

44. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Put d = 7 into 2d + a1 = -3,
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

45. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Put d = 7 into 2d + a1 = -3,
2(7) + a1 = -3
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

46. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Put d = 7 into 2d + a1 = -3,
2(7) + a1 = -3
14 + a1 = -3
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

47. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Put d = 7 into 2d + a1 = -3,
2(7) + a1 = -3
14 + a1 = -3
a1 = -17
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

48. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Put d = 7 into 2d + a1 = -3,
2(7) + a1 = -3
14 + a1 = -3
a1 = -17
Hence the specific formula is an = 7(n – 1) – 17
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

49. Example D. Given that a1, a2 , a3 , …is an arithmetic sequence
with a3 = -3 and a9 = 39, find d, a1 and the specific formula.
Set n = 3 and n = 9 in the general arithmetic formula
an = d(n – 1) + a1, we get
a3 = d(3 – 1) + a1 = -3
2d + a1 = -3
Subtract these equations:
8d + a1 = 39
) 2d + a1 = -3
6d = 42
d = 7
Put d = 7 into 2d + a1 = -3,
2(7) + a1 = -3
14 + a1 = -3
a1 = -17
Hence the specific formula is an = 7(n – 1) – 17
or an = 7n – 24.
Arithmetic Sequences
a9 = d(9 – 1) + a1 = 39
8d + a1 = 39

50. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
Sums of Arithmetic Sequences

51. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
Head Tail
Sums of Arithmetic Sequences

52. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Sums of Arithmetic Sequences

53. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
Sums of Arithmetic Sequences

54. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula.
Sums of Arithmetic Sequences

55. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula. Find d = 7 – 4 = 3.
Sums of Arithmetic Sequences

56. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula. Find d = 7 – 4 = 3.
Therefore the specific formula is
an = 3(n – 1) + 4
Sums of Arithmetic Sequences

57. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula. Find d = 7 – 4 = 3.
Therefore the specific formula is
an = 3(n – 1) + 4
an = 3n + 1.
Sums of Arithmetic Sequences

58. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula. Find d = 7 – 4 = 3.
Therefore the specific formula is
an = 3(n – 1) + 4
an = 3n + 1.
Sums of Arithmetic Sequences
If an = 67 = 3n + 1,

59. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula. Find d = 7 – 4 = 3.
Therefore the specific formula is
an = 3(n – 1) + 4
an = 3n + 1.
Sums of Arithmetic Sequences
If an = 67 = 3n + 1, then
66 = 3n

60. Given that a1, a2 , a3 , …an an arithmetic sequence, then
a1+ a2 + a3 + … + an = n
TailHead +
2
( )
ana1 +
2( )= n
Head Tail
Example E.
a. Given the arithmetic sequence a1= 4, 7, 10, … , and
an = 67. What is n?
We need the specific formula. Find d = 7 – 4 = 3.
Therefore the specific formula is
an = 3(n – 1) + 4
an = 3n + 1.
Sums of Arithmetic Sequences
If an = 67 = 3n + 1, then
66 = 3n
or 22 = n

61. b. Find the sum 4 + 7 + 10 +…+ 67
Sums of Arithmetic Sequences

62. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22,
Sums of Arithmetic Sequences

63. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22
4 + 67
2
( )
Sums of Arithmetic Sequences

64. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22
4 + 67
2
( )
11
Sums of Arithmetic Sequences

65. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22 = 11(71) = 781
4 + 67
2
( )
11
Sums of Arithmetic Sequences

66. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22 = 11(71) = 781
4 + 67
2
( )
11
Sums of Arithmetic Sequences
ana1 +
2( ) =
Formulas for the Arithmetic Sums
The sum Sn of the first n terms of an arithmetic sequence
a1, a2 , a3 , …an, i.e.
a1+ a2 + a3 + … + an = Sn= n
2a1 + (n –1)d
2
( )n

67. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22 = 11(71) = 781
4 + 67
2
( )
11
Sums of Arithmetic Sequences
ana1 +
2( ) =
Formulas for the Arithmetic Sums
The sum Sn of the first n terms of an arithmetic sequence
a1, a2 , a3 , …an, i.e.
a1+ a2 + a3 + … + an = Sn= n
2a1 + (n –1)d
2
( )n
Example F.
a. How many bricks are
there as shown
if there are 100
layers of bricks
continuing in the same pattern?

68. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22 = 11(71) = 781
4 + 67
2
( )
11
Sums of Arithmetic Sequences
ana1 +
2( ) =
Formulas for the Arithmetic Sums
The sum Sn of the first n terms of an arithmetic sequence
a1, a2 , a3 , …an, i.e.
a1+ a2 + a3 + … + an = Sn= n
2a1 + (n –1)d
2
( )n
Example F.
a. How many bricks are
there as shown
if there are 100
layers of bricks
continuing in the same pattern?
The 1st layer has 3 = 1 x 3 bricks the 2nd layer has 6 = 2 x 3
bricks, etc..,

69. b. Find the sum 4 + 7 + 10 +…+ 67
a1 = 4, and a22 = 67 with n = 22, so the sum
4 + 7 + 10 +…+ 67 = 22 = 11(71) = 781
4 + 67
2
( )
11
Sums of Arithmetic Sequences
ana1 +
2( ) =
Formulas for the Arithmetic Sums
The sum Sn of the first n terms of an arithmetic sequence
a1, a2 , a3 , …an, i.e.
a1+ a2 + a3 + … + an = Sn= n
2a1 + (n –1)d
2
( )n
Example F.
a. How many bricks are
there as shown
if there are 100
layers of bricks
continuing in the same pattern?
The 1st layer has 3 = 1 x 3 bricks the 2nd layer has 6 = 2 x 3
bricks, etc.., hence the 100th layer has 100 x 3 = 300 bricks.

70. Sums of Arithmetic Sequences
The 1st layer has 3 bricks

71. Sums of Arithmetic Sequences
The 1st layer has 3 bricks
The last layer has 300 bricks

72. Sums of Arithmetic Sequences
The 1st layer has 3 bricks
n = 100 layers
The last layer has 300 bricks

73. Sums of Arithmetic Sequences
The sum 3 + 6 + 9 + .. + 300 is arithmetic.
The 1st layer has 3 bricks
n = 100 layers
The last layer has 300 bricks

74. Sums of Arithmetic Sequences
3 + 300
2( )
The sum 3 + 6 + 9 + .. + 300 is arithmetic.
Hence the total number of bricks is
The 1st layer has 3 bricks
n = 100 layers
100
The last layer has 300 bricks

75. Sums of Arithmetic Sequences
3 + 300
2( )
The sum 3 + 6 + 9 + .. + 300 is arithmetic.
Hence the total number of bricks is
The 1st layer has 3 bricks
n = 100 layers
100
The last layer has 300 bricks
= 50 x 303
= 15150

76. Arithmetic Sequences
2. –2, –5, –8, –11,..1. 2, 5, 8, 11,..
4. –12, –5, 2, 9,..3. 6, 2, –2, –6,..
6. 23, 37, 51,..5. –12, –25, –38,..
8. –17, .., a7 = 13, ..7. 18, .., a4 = –12, ..
10. a12 = 43, d = 59. a4 = –12, d = 6
12. a42 = 125, d = –511. a8 = 21.3, d = –0.4
14. a22 = 25, a42 = 12513. a6 = 21, a17 = 54
16. a17 = 25, a42 = 12515. a3 = –4, a17 = –11,
Exercise A. For each arithmetic sequence below
a. find the first term a1 and the difference d
b. find a specific formula for an and a100
c. find the sum  ann=1
100

77. B. For each sum below, find the specific formula of
the terms, write the sum in the  notation,
then find the sum.
1. – 4 – 1 + 2 +…+ 302
Sum of Arithmetic Sequences
2. – 4 – 9 – 14 … – 1999
3. 27 + 24 + 21 … – 1992
4. 3 + 9 + 15 … + 111,111,111
5. We see that it’s possible to add infinitely many
numbers and obtain a finite sum.
For example ½ + ¼ + 1/8 + 1/16... = 1.
Give a reason why the sum of infinitely many terms
of an arithmetic sequence is never finite,
except for 0 + 0 + 0 + 0..= 0.

78. Arithmetic Sequences
1. a1 = 2
d = 3
an = 3n – 1
a100 = 299
 an = 15 050
(Answers to the odd problems) Exercise A.
n=1
100
3. a1 = 6
d = – 4
an = – 4n + 10
a100 = – 390
 an = – 19 200n=1
100
5. a1 = – 12
d = –13
an = – 13n + 1
a100 = – 129
 an = – 65 550n=1
100
7. a1 = 18
d = – 10
an = – 10n +28
a100 = – 972
 an = – 47 700n=1
100
9. a1 = –30
d = 6
an = 6n – 36
a100 = 564
 an = 26 700n=1
100
11. a1 = 24.1
d = –0.4
an = –0.4n + 24.5
a100 = –15.5
 an = 430n=1
100

79. Arithmetic Sequences
13. a1 = 6
d = 3
an = 3n + 3
a100 = 303
 an = 15 450n=1
100
15. a1 = –3
d = – 0.5
an = – 0.5n – 2.5
a100 = –52.5
 an = –2 775n=1
100
Exercise B.
1. – 4 – 1 + 2 +…+ 302 =  3n – 7 = 15 347
3. 27 + 24 + 21 … – 1992 =  –3n + 30 = –662 205
n=1
103
n=1
674