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: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
2. Identify the given values: the first term \(a_1 = 1,000,000\), the common ratio \(r = 0.1\), and the term to find is \(a_8\) (the 8th term).
3. Substitute the known values into the formula: \(a_8 = 1,000,000 \times (0.1)^{8-1}\).
4. Simplify the exponent: calculate \$8 - 1\( to get 7, so the expression becomes \)a_8 = 1,000,000 \times (0.1)^7$.
5. Evaluate the power \((0.1)^7\) and then multiply by 1,000,000 to find the value of \(a_8\).

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. For example, if the first term is 2 and the ratio is 3, the sequence is 2, 6, 18, 54, 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 and Powers

Exponentiation involves raising a number to a power, which means multiplying the number by itself a certain number of times. In geometric sequences, powers of the common ratio determine how terms grow or shrink, making understanding exponents essential for calculating terms.