sequences and summation notation

1. Example C.
 fk = f4+ f5+ f6+ f7+ f8k=4
8
 ai = a2+ a3+ a4+ a5i=2
5
 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
 aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
k=5 k=6 k=7 k=8
b.  (–1)k(3k + 2)
=(–1)3(3*3+2)+(–1)4(3*4+2)+(–1)5(3*5+2)
= –11 + 14 – 17 = –14
k=3
5
Example D.

2. Properties Summation Notation
 k =
Formula for the Sum of Natural Numbers
k=1
n
n(n + 1)
2
k=1
45
 (2k – 5) = Σ2k – Σ5 = 2Σk – Σ5k=1
45
k k=1
45
= 2 – 5*4545(45 + 1)
2
Example E. Find  (2k – 5)
k
45
k=1
= 2070 – 225 = 1845

3. Exercise A. List the first four terms of each of the
following sequences given by fn where n = 1,2, 3, ..
Sequences
2.
4. 5. 6.
7. 8.
9. 10.
1. 3.fn = –5 + n fn = 5 – n fn = 3n
fn = –5 + 2n fn = 5 – n2fn = –4n + 1
fn = (–1)n5 / n
fn = (3n + 2)/(–1 – n)
fn = 2n2 – n
fn = n2 / (2n + 1)
B. Find formulas fn for the following sequences.
2.
4.
5. 6.
7. 8.
9.
10.
1.
3.
2, 3, 4, 5.. –3, –2, –1, 0, 1..
10, 20, 30, 40,.. 5, 10, 15, 20,..
–40, –30, –20, –10, 0,.. –5, –10, –15, –20,..
1/2, 1/3, 1/4, 1/5.. 1/2, –2/3, 3/4, –4/5..
–1, 1/4, –1/9, 1/16, –1/25,..
1, 0.1, 0.001, 0.0001,..

4. Sequences

5. Sequences
It’s possible to add infinitely many numbers and obtain a
finite sum. For example, the sum ½ + ¼ + 1/8 + 1/16...
represents the accumulated amount of
“taking half of the 1 or ½,
take half of what’s left, or ¼,
then take of half of what’s left or 1/8,
and repeat the process without stopping..”
We see that ½ + ¼ + 1/8 + 1/16 + 1/32... = 1.
..
= 1
7. What is 1/3 + 1/9 + 1/27 + 1/81... = ?
½
¼
1/8
1/16
1/32
(Hint: Let the sum 1/3 + 1/9 + 1/27 + 1/81... = x,
factoring out 1/3 from the left, we’ve
1/3(1 + 1/3 + 1/9 + 1/27 + 1/81...) = x, or
1/3(1 + x) = x, then solve for x.)
8. What is 1/4 + 1/16 + 1/64 + 1/81... = ? (Hint: factor out ¼)
9. What is 1/5 + 1/25 + 1/125 + 1/625... = ? (Hint: factor out 1/5)

6. (Answers to the odd problems) Exercise A.
1. f1 = –4, f2 = –3, f3 = –2, f4 = –1
3. f1 = 3, f2 = 6, f3 = 9, f4 = 12
5. f1 = –3, f2 = –7, f3 = –13, f4 = – 15
9. f1 = -5/2, f2 = -8/3, f3 = -11/4, f4 = -14/5
Exercise B.
1. fn= n+1 3. fn= 10n 5. fn= –10(5–n)
7. fn= 1/(n+1) 9. fn= (–1)n/n2
Exercise C.
1. 16 3. 26 5. 70
9.–50
7. 90
11. 436 13. 11/6 15. 100/101
Sequences
7. f1 = 1, f2 = 6, f3 = 15, f4 = 28
Exercise D. 1. 6(𝑘 + 1) 3. 9𝑗 − 3 5. 21(𝑘 + 3)
7. 1/2 9. 1/4