In Exercises 1–8, write the first five terms of each geometric sequence. a₁ = 5, r = 3

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 $a{1}_{}=\; 5$ and the common ratio $r=\; 3$.
3. Calculate the second term using the formula: $a{2}_{}=\; 5\; \backslash times\; 3^\{2-1\}\; =\; 5\; \backslash times\; 3$.
4. Calculate the third, fourth, and fifth terms similarly by substituting $n\; =\; 3,\; 4,\; 5$ into the formula: $a{n}_{}=\; 5\; \backslash times\; 3^\{n-1\}$.
5. List the first five terms in order: $a\_1,\; a\_2,\; a\_3,\; a\_4,\; a\_5$ using the values calculated 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 remains the same throughout the sequence, creating a consistent pattern of growth or decay.

Common Ratio (r)

The common ratio is the fixed factor by which each term in a geometric sequence is multiplied to get the next term. In this problem, the ratio is 3, meaning each term is three times the previous term.

Finding Terms of a Geometric Sequence

To find terms in a geometric sequence, use 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 helps calculate any term without listing all previous terms.