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High School Mathematics - 2
3.10 Formula for nth term in a Sequence

Example:
Find the formula for nth term in the sequence: 2, 5, 8, 11, 14, . . .
This is an arithmetic sequence
d = 3
a1 = 2
an = 2 + (n - 1)3
Therefore : The nthth term for the sequence is an = 3n -1
Verify: By putting in 1, 2, 3, ...... and you get the sequence

Example:
Find the formula for nth term in the sequence: 2, 4, 8, 16, 32, . . .
This is geometric sequence.
a1 = 2
r = 2
Therefore, an = 2 . 2(n - 1)
an = 2.2n-1
Verify: By putting in 1, 2, 3, ........and see if you get the sequence

Example:
Find the formula for nth term in the sequence: 21, 201, 2001, 20001, . . .
This is neither geometric or arithmetic.
Think of the sequence as (20 +1), (200+1), (2000 + 1), (20000 + 1), . . .
It can be written as the sequence:[(2)(10) +1],[(2)(100) +1], [(2)(1000) +1], [(2)(10000) +1]
Check and see a pattern with powers of 10
an = 2.10n + 1

Example:
Find the formula for nth term in the sequence: 3, 6, 18, 72, . . .
This is neither arithmetic nor geometric sequence
The ratio of second to first term is 6/3 = 2
The ratio of third to second term is 18/6 = 3
The ratio of fourth to third term is 72/18 =4
3, 6, 18, 72, . . . . .
=3(1, 2, 3, 4, . . . . )
=3n!
Therefore : The nthth term for the sequence is 3n!

More examples:
Arithmetic Series:
Seriesnth term
1,2,3,4,5.........n
2,4,6.........2n
3,5,7.........2n+1
7,10,13,16.........3n+4
5,10,15.........5n
-10, -5, 0, 5........5n-15
Geometic Series:
Seriesnth term
5, 10, 20, 40,....5 X 2n-1
21, 22,23, 24,... 2n
51, 52, 53, 54,... 5n
31, 32, 33, 34,... 3n
7x21, 7x22, 7x23, 7x24,... 7x2n
Other:
Seriesnth term
1, 4, 9, 16.........
12, 22, 32.........
n2
1, 8, 27, 64.........
13, 23, 33.........
n3
4x12, 4x22, 4x32,.....4xn2
1,2,6,24,120......
1, 2x1 ,3x2x1, 4x3x2x1, 5x4x3x2x1......
n!
1,-1,1,-1,1........ (-1)n+1


Directions: Find the formula for the nth term in the sequence. Also write at least 5 examples of your own.
Q 1: Find a formula for the nth term of the geometric sequence: 7x21, 7x22, 7x23, 7x24,...
2n-7
7n-1
7x2n

Q 2: Find a formula for the nth term of the arithmetic sequence: 10, -5, 0, 5......
15 - 5n
5n - 15
15 + 5n

Q 3: Find a formula for the nth term of the geometric sequence: 5, 10, 20, 40,....
5n-1
5 X 21-n
5 x 2n-1

Q 4: Find the formula for the nth term in the arithmetic sequence: -10, -5, 0, 5, ....
n + 1
15 + 5n
5n - 15

Q 5: Find the formula for the nth term in the arithmetic sequence: 10, 9, 8, 7, 6,......
11 - n
11 + n
9 + n

Q 6: Find a formula for the nth term of the arithmetic sequence: -2, -8, -14, -20, .....
4n + 6
4 - 6n
4n - 6

Q 7: Find the formula for the nth term in the arithmetic sequence: 3, 7, 11, 15, 19, . . .
an = 4n - 1
an = an - 1 + 1
an = 3n - 1

Q 8: Find the formula for the nth term in the arithmetic sequence: 31, 32, 33, 34,...
n3
3n
2n

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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