Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a(sub n) to find a(sub 8), the eighth term of the sequence. 1, 2, 4, 8, ...

Verified step by step guidance

Step 1: Recognize that the sequence provided is geometric because each term is obtained by multiplying the previous term by a constant factor. In this case, the common ratio (r) can be calculated as r = 2/1 = 2.

Step 2: Recall the general formula for the nth term of a geometric sequence:

a

_n = a₁ ⋅ r^n-1, where a₁ is the first term, r is the common ratio, and n is the term number.

Step 3: Substitute the known values into the formula. Here, a₁ = 1 and r = 2. The formula becomes:

a

_n = 1 ⋅ 2^n-1.

Step 4: To find the eighth term (a₈), substitute n = 8 into the formula:

a

₈ = 1 ⋅ 2^8-1.

Step 5: Simplify the exponent in the formula to calculate a₈. This involves evaluating

2

⁷, which is the eighth term of the 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. For example, in the sequence 1, 2, 4, 8, the common ratio is 2, as each term is obtained by multiplying the previous term by 2.

General Term Formula

The general term (nth term) of a geometric sequence can be expressed using the formula a(n) = a(1) * r^(n-1), where a(1) is the first term, r is the common ratio, and n is the term number. This formula allows you to calculate any term in the sequence based on its position.

Finding Specific Terms

To find a specific term in a geometric sequence, you substitute the desired term number into the general term formula. For instance, to find the eighth term (a(8)) of the sequence 1, 2, 4, 8, you would use the formula a(8) = 1 * 2^(8-1), which simplifies to 128.