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

A geometric sequence has a first term of $\frac{1}{2}$ and a common ratio of $\frac{1}{4}$ . What is the sum of the first 3 terms of this sequence?

A
$\frac{1}{14}$
B
$\frac{13}{16}$
$\frac{21}{32}$ Answer
D
$\frac{1}{32}$
E
$\frac{9}{128}$

Answer

The sum of the first 3 terms of this sequence is

\frac{21}{32}

The sum of the first three terms of a geometric sequence is calculated by finding each individual term and then adding them together. The first term is

\frac{1}{2}

. The second term is obtained by multiplying the first term by the common ratio:

\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}

. The third term is obtained by multiplying the second term by the common ratio:

\frac{1}{8} \times \frac{1}{4} = \frac{1}{32}

. To add these terms, we find a common denominator of 32:

\frac{16}{32} + \frac{4}{32} + \frac{1}{32} = \frac{21}{32}

Step-by-Step Solution

Identify the first three terms of the geometric sequence using the formula

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

The first term

a_1

is given as

\frac{1}{2}

. The second term is

a_2 = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}

. The third term is

a_3 = \frac{1}{8} \cdot \frac{1}{4} = \frac{1}{32}

Before calculating the sum, each individual term to be summed must be determined.

Find a common denominator to add the three fractional terms.

The least common multiple of the denominators 2, 8, and 32 is 32. Express the terms with this common denominator:

a_1 = \frac{16}{32}

a_2 = \frac{4}{32}

, and

a_3 = \frac{1}{32}

Adding fractions requires a common denominator.

Sum the adjusted fractions.

\frac{16}{32} + \frac{4}{32} + \frac{1}{32} = \frac{21}{32}

This yields the total sum of the first three terms.

Key Concept

Calculating the sum of a finite geometric series by finding and summing individual terms.

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