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### High School Mathematics - 23.10 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: Series nth 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: Series nth 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: Series nth 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 the formula for the nth term in the arithmetic sequence: -10, -5, 0, 5, .... 15 + 5n5n - 15n + 1 Q 2: Find the formula for the nth term in the arithmetic sequence: 10, 9, 8, 7, 6,......11 + n11 - n9 + n Q 3: Find the formula for the nth term in the arithmetic sequence: 31, 32, 33, 34,...3nn32n Q 4: Find a formula for the nth term of the geometric sequence: 101, 102, 103, 104,........ 10n-110nn10 Q 5: Find a formula for the nth term of the geometric sequence: 5, 10, 20, 40,....5 X 21-n5n-15 x 2n-1 Q 6: Find the formula for the nth term in the arithmetic sequence: 3, 7, 11, 15, 19, . . . an = 3n - 1an = an - 1 + 1an = 4n - 1 Q 7: Find a formula for the nth term of the geometric sequence: 1, 0.5, 0.25, 0.125, ........2n-1n1-n21-n Q 8: Find a formula for the nth term of the arithmetic sequence: -2, -8, -14, -20, .....4n + 64 - 6n4n - 6 Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!