FINDING THE GENERAL TERM 10

1. FINDING THE
GENERAL TERM
THE COMMON DIFFERENCE IS NOT CONSTANT

2. EXAMPLE:
FIND THE GENERAL TERM OF
THE SEQUENCE
1, 3, 6, 10, 15, . . .

3. Prepare a table
n 1 2 3 4 5 . . . n
𝒂 𝒏
1 3 6 10 15 . . . ?

4. Get the difference
𝟏, 𝟑, 𝟔, 𝟏𝟎, 𝟏𝟓, . . .
𝒂𝒏 𝟐
+ 𝒃𝒏 + 𝒄 = 𝒂 𝒏
+𝟐 +𝟑 +𝟒 +𝟓
+𝟏

5. Solve for a, b and c.
𝑰𝒇 𝒏 = 𝟏 𝒂𝒏𝒅 𝒂 𝒏 = 𝟏
𝑰𝒇 𝒏 = 𝟐 𝒂𝒏𝒅 𝒂 𝒏 = 𝟑
𝒂𝒏 𝟐
+ 𝒃𝒏 + 𝒄 = 𝒂 𝒏
𝒂(𝟏) 𝟐
+ 𝒃(𝟏) + 𝒄 = 𝟏
𝒂 + 𝒃 + 𝒄 = 𝟏 𝑬𝒒. 𝟏
𝒂(𝟐) 𝟐
+ 𝒃(𝟐) + 𝒄 = 𝟑
𝟒𝒂 + 𝟐𝒃 + 𝒄 = 𝟑 𝑬𝒒. 𝟐

6. Solve for a, b and c.
𝑰𝒇 𝒏 = 𝟑 𝒂𝒏𝒅 𝒂 𝒏 = 𝟔
𝒂𝒏 𝟐
+ 𝒃𝒏 + 𝒄 = 𝒂 𝒏
𝒂(𝟑) 𝟐 + 𝒃(𝟑) + 𝒄 = 𝟔
𝟗𝒂 + 𝟑𝒃 + 𝒄 = 𝟔 𝑬𝒒. 𝟑

7. 𝑺𝒖𝒃𝒕𝒓𝒂𝒄𝒕 𝑬𝒒. 𝟏 𝒇𝒓𝒐𝒎 𝑬𝒒. 𝟐
Solve for a, b and c.
𝒂 + 𝒃 + 𝒄 = 𝟏 𝑬𝒒. 𝟏
𝟒𝒂 + 𝟐𝒃 + 𝒄 = 𝟑 𝑬𝒒. 𝟐
𝟑𝒂 + 𝒃 = 𝟐 𝑬𝒒. 𝟒
𝑺𝒖𝒃𝒕𝒓𝒂𝒄𝒕 𝑬𝒒. 𝟐 𝒇𝒓𝒐𝒎 𝑬𝒒. 𝟑
𝟒𝒂 + 𝟐𝒃 + 𝒄 = 𝟑 𝑬𝒒. 𝟐
𝟗𝒂 + 𝟑𝒃 + 𝒄 = 𝟔 𝑬𝒒. 𝟑
𝟓𝒂 + 𝒃 = 𝟑 𝑬𝒒. 𝟓

8. 𝑺𝒖𝒃𝒕𝒓𝒂𝒄𝒕 𝑬𝒒. 𝟒 𝒇𝒓𝒐𝒎 𝑬𝒒. 𝟓
𝟑𝒂 + 𝒃 = 𝟐 𝑬𝒒. 𝟒
𝟓𝒂 + 𝒃 = 𝟑 𝑬𝒒. 𝟓
𝒂 =
𝟏
𝟐
𝟐𝒂 = 𝟏
𝑹𝒆𝒑𝒍𝒂𝒄𝒆 𝒕𝒉𝒆 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂 𝒕𝒐 𝒕𝒉𝒆 𝑬𝒒. 𝟒
𝟑𝒂 + 𝒃 = 𝟐 𝑬𝒒. 𝟒
𝟑(
𝟏
𝟐
) + 𝒃 = 𝟐
𝟑
𝟐
+ 𝒃 = 𝟐
𝒃 = 𝟐 −
𝟑
𝟐
𝒃 =
𝟏
𝟐

9. 𝒄 = 𝟎
𝑹𝒆𝒑𝒍𝒂𝒄𝒆 𝒕𝒉𝒆 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂 𝒂𝒏𝒅 𝒃 𝒊𝒏 𝒕𝒉𝒆 𝑬𝒒. 𝟏
𝒂 + 𝒃 + 𝒄 = 𝟏 𝑬𝒒. 𝟏
𝟏
𝟐
+
𝟏
𝟐
+ 𝒄 = 𝟏
𝟏 + 𝒄 = 𝟏
𝒄 = 𝟏 − 𝟏

10. 𝒂𝒏 𝟐
+ 𝒃𝒏 + 𝒄 = 𝒂 𝒏
𝟏
𝟐
𝒏 𝟐
+
𝟏
𝟐
𝒏 + 𝟎 = 𝒂 𝒏
𝒂 𝒏 =
𝟏
𝟐
𝒏(𝒏 + 𝟏)
𝑻𝒉𝒆𝒓𝒆𝒇𝒐𝒓𝒆 𝒕𝒉𝒆
𝑮𝑬𝑵𝑬𝑹𝑨𝑳 𝑻𝑬𝑹𝑴
𝒐𝒇 𝒕𝒉𝒆 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆
𝟏, 𝟑, 𝟔, 𝟏𝟎, 𝟏𝟓, . . . 𝒊𝒔

11. QUI
Z
In your notebook answer the
following.
1. FIND THE
GENERAL TERM OF
THE SEQUENCE
1, 6, 15, 28, 45, . . .
2. FIND THE
GENERAL TERM
OF THE
SEQUENCE
1, 7, 18, 34, 55, . . .
To be submitted TODAY (SEPTEMBER 23, 2020)
until 3pm only.

12. ASSIGNMENT In your notebook answer the
following.
1. FIND THE GENERAL
TERM OF THE
SEQUENCE
2, 5, 10, 17, 26, . . .
2. FIND THE GENERAL
TERM OF THE
SEQUENCE
2, 6, 12, 20, 30, . . .
3. FIND THE GENERAL
TERM OF THE
SEQUENCE
5, 7, 11, 17, 25, . . .
To be submitted on
THURSDAY (OCTOBER
1, 2020) until 4pm
only.

13. SERIES

14. Series
It is the sum of the terms
of a sequence.

15. EXAMPLE
1, 3, 5, 7, 9, 11, . . .
1+3+5+7+9+11+, . . .
SEQUENCE
SERIES
𝑺 𝟔
𝒔𝒖𝒎 𝒐𝒇
𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕
𝟔 𝒕𝒆𝒓𝒎𝒔

16. EXAMPLE
In the sequence 1, 3, 6, 10, 15, . . . We have the
following partial sums:
𝑺 𝟏 = 𝟏 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒕𝒆𝒓𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆
𝑺 𝟐 = 𝟏 + 𝟑 = 𝟒
𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝟐 𝒕𝒆𝒓𝒎𝒔

17. EXAMPLE
In the sequence 1, 3, 6, 10, 15, . . . We have the
following partial sums:
𝑺 𝟑 = 𝟏 + 𝟑 + 𝟔 =
𝑺 𝟒 = 𝟏 + 𝟑 + 𝟔 + 𝟏𝟎 = 𝟐𝟎
𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝟒 𝒕𝒆𝒓𝒎𝒔
𝟏𝟎
𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝟑 𝒕𝒆𝒓𝒎𝒔

18. EXAMPLE
In the sequence 1, 3, 6, 10, 15, . . . We have the
following partial sums:
𝑺 𝟓 = 𝟏 + 𝟑 + 𝟔 + 𝟏𝟎 + 𝟏𝟓 = 𝟑𝟓
𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝟓 𝒕𝒆𝒓𝒎𝒔

19. EXAMPLE
5, 15, 25, 35, 45.
𝑺 𝟐 =
𝑺 𝟐 = 𝟓 + 𝟏𝟓 = 𝟐𝟎
𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝟐 𝒕𝒆𝒓𝒎𝒔
𝑺 𝟒 = 8𝟎 𝑺 𝟓 = 𝟏𝟐𝟓

20. EXAMPLE
2, -6, 18, -54, 162, -486, . . .
𝑺 𝟒 =
𝑺 𝟔 = −𝟑𝟔𝟒
−𝟒𝟎

21. Sigma Notation
It tells us to sum or add up
terms.
It is the other way to
express the sum of terms.
Σ
(sigma)

22. Start (lower limit)
𝒌
𝒏
𝟑𝒌
Index of summation
end (upper limit)

23. EXAMPLE: Express each sum using
sigma notation.
𝟏 +
𝟏
𝟐
+
𝟏
𝟑
+
𝟏
𝟒
+
𝟏
𝟓 𝒌=𝟏
𝟓
𝟏
𝒌
𝟏 𝟐
+ 𝟐 𝟐
+ 𝟑 𝟐
+. . . +𝒏 𝟐
𝒌=𝟏
𝒏
𝒌 𝟐

24. ASSIGNMEN
T
In your book answer
page 21
I. Practice and
application (even
numbers only)
IV. # 36, 37 and 38.
To be
submitted on
THURSDAY
(OCTOBER 2,
2020) until
3pm only.