9. Sequences, Series, & Induction

Geometric Sequences

9. Sequences, Series, & Induction

Geometric Sequences

2:30 minutes

Problem 36

Textbook Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 6) when a(sub 1) = 16, r = 1/2

Verified step by step guidance

Step 1: Recall the formula for the nth term of a geometric sequence:

a

_n = a₁ ⋅ r^n-1, where

a

₁ is the first term,

r

is the common ratio, and

n

is the term number.

Step 2: Substitute the given values into the formula. Here,

a

₁ = 16,

r = 1/2

, and

n = 6

. The formula becomes:

a

₆ = 16 ⋅ (1/2)^6-1.

Step 3: Simplify the exponent in the formula. Since

6 - 1 = 5

, the formula becomes:

a

₆ = 16 ⋅ (1/2)⁵.

Step 4: Evaluate the power of the common ratio. Calculate

(1/2)

⁵, which represents multiplying

1/2

by itself five times.

Step 5: Multiply the result of

(1/2)

⁵ by

16

to find

a

₆. This will give the sixth term of the geometric sequence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of sequence can be expressed in the form a(n) = a(1) * r^(n-1), where a(n) is the nth term, a(1) is the first term, r is the common ratio, and n is the term number.

General Term Formula

The general term formula for a geometric sequence allows us to calculate any term in the sequence based on its position. Specifically, the nth term can be calculated using the formula a(n) = a(1) * r^(n-1). This formula is essential for finding specific terms in the sequence, such as a(sub 6) in this case.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is denoted as 'r' in the general term formula. In the given problem, the common ratio is 1/2, indicating that each term is half of the previous term, which significantly influences the value of the terms in the sequence.

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