Geometric Sequence Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
A geometric sequence has a1=5a_1 = 5 and r=2r = 2. Find a8a_8 and S8S_8.

Solution

  1. 1
    Use the geometric sequence formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} with a1=5a_1 = 5, r=2r = 2, n=8n = 8.
  2. 2
    Find the 8th term: a8=5⋅27=5⋅128=640a_8 = 5 \cdot 2^7 = 5 \cdot 128 = 640
  3. 3
    Apply the partial sum formula: S8=a1⋅1−r81−r=5⋅1−2561−2=5⋅255=1275S_8 = a_1 \cdot \frac{1-r^8}{1-r} = 5 \cdot \frac{1-256}{1-2} = 5 \cdot 255 = 1275

Answer

a8=640a_8 = 640; S8=1275S_8 = 1275
Geometric sequences grow exponentially. The formula an=a1rn−1a_n = a_1 r^{n-1} uses exponent n−1n-1 because the first term has no multiplications applied.

About Geometric Sequence

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio rr.

Learn more about Geometric Sequence →

More Geometric Sequence Examples

Example 2 medium

Bacteria double every 3 hours. Starting with 500, how many after 15 hours?

Example 3 easy

Find [formula] and [formula] for the sequence [formula]

Example 4 hard

In a geometric sequence, [formula] and [formula]. Find [formula] and [formula].

Example 5 medium

Find the 6th term of the geometric sequence: 2, 6, 18, ...

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