In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a₁ and common ratio, r. Find a₈ when a₁ = 6, r = 2

Verified step by step guidance

1. Recall the formula for the nth term of a geometric sequence: $a{n}_{}=a{1}_{}{r}^{\mathrm{n-1}}$, where $a{1}_{}$ is the first term, $r$ is the common ratio, and $n$ is the term number.
2. Identify the given values: $a{1}_{}=\; 6$, $r=\; 2$, and the term to find is $a{8}_{}$ (so $n=\; 8$).
3. Substitute the known values into the formula: $a{8}_{}=\; 6\; \backslash times\; 2^\{8-1\}$.
4. Simplify the exponent expression: calculate $8\; -\; 1\; =\; 7$, so the formula becomes $a{8}_{}=\; 6\; \backslash times\; 2^\{7\}$.
5. To find $a{8}_{}$, multiply 6 by $2^\{7\}$. (You can calculate $2^\{7\}$ as 2 multiplied by itself 7 times.)

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, if the first term is 6 and the ratio is 2, the sequence is 6, 12, 24, and so on.

General Term Formula of a Geometric Sequence

The nth term of a geometric sequence can be found 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 direct calculation of any term without listing all previous terms.

Exponentiation in Sequences

Exponentiation involves raising the common ratio to a power based on the term number minus one. Understanding how to compute powers, such as 2^(8-1) = 2^7, is essential to correctly apply the general term formula and find the desired term in the sequence.