Geometric Sequence

A geometric sequence or geometric progression is a sequence of numbers such that the ratios between successive terms is a constant r, called the common ratio.

Common Ratio Formula

3, 6, 12, 24, 48, ...

6/3 = 2.

12/6 = 2.

24/12 = 2.

48/24 = 2.

r = 2.


Nth Term of a Geometric Sequence

1If the 1st term is known.

an = a1 · rn-1

3, 6, 12, 24, 48, ..

an = 3 · 2n-1 = 3 · 2n · 2-1 = (3/2)· 2n

2If the value that occupies any other term of the sequence is known.

an = ak · rn-k

a4= 24, k=4 and r=2.

an = a4 · rn-4

an = 24· 2n-4= (24/16)· 2n = (3/2) · 2n


Geometric Series

A geometric series is the sum of a geometric sequence.

Compute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...


Infinite Geometric Series

If −1 < r < 1 the infinite geometric series converges to a specific value:

Calculate the sum of the terms of the sequence:


Product

Calculate the product of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

Product of a Sequence Example