How to Find the Nth Term and Sum of a Geometric Sequence: A Step-by-Step Guide with Example

The Geometric Sequence Finite Sum

In a geometric sequence, or geometric progression, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r)

In a geometric sequence, or geometric progression, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). The formula to find the nth term (an) of a geometric sequence is given by:

an = a1 * (r^(n-1))

Where:
an = the nth term of the geometric sequence
a1 = the first term of the geometric sequence
r = the common ratio of the geometric sequence
n = the position of the term in the sequence

The finite sum, or the sum of a specific number of terms, in a geometric sequence can be determined using the formula for the sum (Sn) given by:

Sn = a1 * (1 – r^n) / (1 – r)

Where:
Sn = the sum of the first n terms of the geometric sequence

Let’s illustrate this with an example:

Example:
Consider a geometric sequence with a first term (a1) of 3 and a common ratio (r) of 2. We want to find the sum of the first 5 terms (n = 5) of this sequence.

Using the formula for the nth term:
a5 = a1 * (r^(5-1))
a5 = 3 * (2^4)
a5 = 3 * 16
a5 = 48

Now, using the formula for the finite sum:
S5 = a1 * (1 – r^5) / (1 – r)
S5 = 3 * (1 – 2^5) / (1 – 2)
S5 = 3 * (1 – 32) / (1 – 2)
S5 = 3 * (-31) / (-1)
S5 = 93

Therefore, the sum of the first 5 terms of the geometric sequence with a first term of 3 and a common ratio of 2 is 93.

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