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 = 1 000 000, r = 0.1

Verified step by step guidance

1. Recall the formula for the nth term of a geometric sequence: $a{n}_{}=a{1}_{}{r}^{n-\; 1}$, where $a{1}_{}$ is the first term, $r$ is the common ratio, and $n$ is the term number.
2. Identify the given values: $a{1}_{}=\; 1,000,000$, $r=\; 0.1$, and $n=\; 8$.
3. Substitute the known values into the formula: $a{8}_{}=\; 1,000,000\; \backslash times\; (0.1)^\{8\; -\; 1\}$.
4. Simplify the exponent expression: $8\; -\; 1\; =\; 7$, so the formula becomes $a{8}_{}=\; 1,000,000\; \backslash times\; (0.1)^7$.
5. Calculate $(0.1)^7$ and then multiply by 1,000,000 to find $a{8}_{}$.

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. This ratio can be any real number, and the sequence can increase or decrease depending on its value.

General Term Formula of a Geometric Sequence

The nth term of a geometric sequence is given by 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 and Powers

Exponentiation involves raising a number to a power, which means multiplying the number by itself repeatedly. In geometric sequences, powers of the common ratio determine how the terms grow or shrink as n increases.