Write the first five terms of each geometric sequence. a₁ = 3, r = 2

Verified step by step guidance

1. Identify the first term \(a_1\) and the common ratio \(r\) of the geometric sequence. Here, \(a_1 = 3\) and \(r = 2\).
2. Recall the formula for the \(n\)-th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
3. Calculate the second term \(a_2\) by substituting \(n=2\) into the formula: \(a_2 = 3 \times 2^{2-1} = 3 \times 2\).
4. Calculate the third term \(a_3\) by substituting \(n=3\): \(a_3 = 3 \times 2^{3-1} = 3 \times 2^2\).
5. Continue this process to find the fourth and fifth terms: \(a_4 = 3 \times 2^{4-1}\) and \(a_5 = 3 \times 2^{5-1}\).

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 2, meaning each term is twice the previous term, which determines how the sequence progresses.

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. Applying this formula allows you to calculate any term, including the first five terms as requested.