A geometric sequence has a first term of \frac{1}{4} and a common ratio of \frac{1}{2}. Wh… — ACT

A geometric sequence has a first term of $\frac{1}{4}$ and a common ratio of $\frac{1}{2}$ . What is the 4th term of this sequence?

\frac{1}{32}

The correct answer is found by substituting

a_1 = \frac{1}{4}

r = \frac{1}{2}

, and

n = 4

into the geometric sequence formula

a_n = a_1 \cdot r^{n-1}

. This gives

a_4 = \frac{1}{4} \cdot (\frac{1}{2})^3 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

Identify the given components of the geometric sequence.

The first term is

a_1 = \frac{1}{4}

and the common ratio is

r = \frac{1}{2}

. We want to find the 4th term, so

n = 4

Before performing any calculations, we must identify the values of the variables needed for the geometric sequence formula.

Substitute the values into the general formula for the

n

-th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

a_4 = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^{4-1} = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^3

This formula relates the

n

-th term to the first term and the common ratio.

Evaluate the exponent first, and then multiply the fractions.

a_4 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

Following the order of operations, we raise the common ratio to the 3rd power before multiplying by the first term.

Finding the

n

-th term of a geometric sequence using the general formula

a_n = a_1 \cdot r^{n-1}

Question