Write the first five terms of each geometric sequence. a₁ = 20, r = 1/2

Verified step by step guidance

1. Identify the first term \(a_1\) and the common ratio \(r\) of the geometric sequence. Here, \(a_1 = 20\) and \(r = \frac{1}{2}\).
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: \(a_2 = 20 \times \left(\frac{1}{2}\right)^{2-1}\).
4. Calculate the third term \(a_3\) by substituting \(n=3\): \(a_3 = 20 \times \left(\frac{1}{2}\right)^{3-1}\).
5. Continue this process to find the fourth and fifth terms, \(a_4\) and \(a_5\), by substituting \(n=4\) and \(n=5\) respectively into the formula.

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 20 and the ratio is 1/2, the sequence progresses by repeatedly multiplying by 1/2.

Common Ratio (r)

The common ratio is the fixed factor between consecutive terms in a geometric sequence. It determines how the sequence grows or shrinks. In this problem, the ratio is 1/2, meaning 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. Applying this formula allows you to calculate any term, including the first five terms as requested.