In Exercises 1–8, write the first five terms of each geometric sequence. a₁ = 20, r = 1/2

Verified step by step guidance

1. Recall that a geometric sequence is defined by the formula $a{n}_{}=a{1}_{}{r}^{\mathrm{n-1}}$, where $a{1}_{}$ is the first term and $r$ is the common ratio.
2. Identify the given values: the first term $a{1}_{}=\; 20$ and the common ratio $r=\; \backslash frac\{1\}\{2\}$.
3. Calculate the second term using the formula: $a{2}_{}=\; 20\; \backslash times\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{2-1\}\; =\; 20\; \backslash times\; \backslash frac\{1\}\{2\}$.
4. Calculate the third, fourth, and fifth terms similarly by increasing the exponent on $r$ by 1 each time: $a\_3\; =\; 20\; \backslash times\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^2$, $a\_4\; =\; 20\; \backslash times\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^3$, and $a\_5\; =\; 20\; \backslash times\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^4$.
5. List the first five terms as $a\_1,\; a\_2,\; a\_3,\; a\_4,\; a\_5$ using the expressions found in the previous steps.

Key Concepts

Geometric Sequence Definition

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 it determines how the sequence progresses.

Common Ratio (r)

The common ratio is the fixed factor between consecutive terms in a geometric sequence. It is denoted by r, and each term is obtained by multiplying the previous term by r. For example, if r = 1/2, each term is half the previous term.

Finding Terms of a Geometric Sequence

To find the nth term of a geometric sequence, use the formula a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio. This formula helps generate any term in the sequence, including the first five terms.