A geometric sequence consists of positive terms and has a first term of 12 and a common ra… — ACT

A geometric sequence consists of positive terms and has a first term of $12$ and a common ratio of $r$ . An arithmetic sequence has a first term of $5$ and a common difference of $d$ . If the $3\text{rd}$ term of the geometric sequence is $3$ and the $4\text{th}$ term of the arithmetic sequence is $7$ , what is the value of $r + d$ ?

A
$\frac{3}{5}$
B
$\frac{11}{8}$
$\frac{7}{6}$ Answer
D
$-\frac{15}{2}$
E
$\frac{35}{48}$

Answer

\frac{7}{6}

The correct value is

\frac{7}{6}

. This is found by first calculating the common ratio of the geometric sequence, where

12r^2 = 3 \Rightarrow r^2 = \frac{1}{4} \Rightarrow r = \frac{1}{2}

, and the common difference of the arithmetic sequence, where

5 + 3d = 7 \Rightarrow 3d = 2 \Rightarrow d = \frac{2}{3}

. Adding these two fractions with a common denominator yields

\frac{3}{6} + \frac{4}{6} = \frac{7}{6}

Step-by-Step Solution

Find the common ratio

r

of the geometric sequence.

r = \frac{1}{2}

The formula for the

n\text{th}

term of a geometric sequence is

g_n = g_1 \cdot r^{n-1}

. For the

3\text{rd}

term,

g_3 = 12r^2 = 3

, which simplifies to

r^2 = \frac{1}{4}

. Since the sequence has positive terms, we take the positive square root to get

r = \frac{1}{2}

Find the common difference

d

of the arithmetic sequence.

d = \frac{2}{3}

The formula for the

n\text{th}

term of an arithmetic sequence is

a_n = a_1 + (n-1)d

. For the

4\text{th}

term,

a_4 = 5 + 3d = 7

, which simplifies to

3d = 2

, or

d = \frac{2}{3}

Calculate the sum of

r

and

d

r + d = \frac{7}{6}

Adding the two values with a common denominator of

6

gives

\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}

Key Concept

Arithmetic and Geometric Sequences and Series

Estimated Time:1m 30s

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