Two infinite geometric series, Series A and Series B, are defined as follows: * Series A h… — ACT

Two infinite geometric series, Series A and Series B, are defined as follows:

* Series A has a first term of $2^x$ and a common ratio of $\frac{1}{2}$ .
* Series B has a first term of $2^{x+3}$ and a common ratio of $\frac{3}{4}$ .

If the sum of Series A and Series B is 136, what is the value of $x$ ?

A
$0.5$
B
$1$
$2$ Answer
D
$3$
E
$4$

Answer

The correct answer is the value 2. Substituting

x = 2

into the sum of Series A gives

2^{2+1} = 8

. Substituting

x = 2

into the sum of Series B gives

2^{2+5} = 128

. The total sum is

8 + 128 = 136

, which matches the given sum.

Step-by-Step Solution

Express the sum of Series A, denoted as

S_A

, using the infinite geometric series sum formula

S = \frac{a_1}{1-r}

S_A = \frac{2^x}{1 - 1/2} = \frac{2^x}{1/2} = 2 \cdot 2^x = 2^{x+1}

Since the common ratio

r = 1/2

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent properties.

Express the sum of Series B, denoted as

S_B

, using the infinite geometric series sum formula.

S_B = \frac{2^{x+3}}{1 - 3/4} = \frac{2^{x+3}}{1/4} = 4 \cdot 2^{x+3} = 2^2 \cdot 2^{x+3} = 2^{x+5}

Since the common ratio

r = 3/4

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent rules.

Set up the equation for the sum of both series and solve for

x

2^{x+1} + 2^{x+5} = 136 \implies 2 \cdot 2^x + 32 \cdot 2^x = 136 \implies 34 \cdot 2^x = 136 \implies 2^x = 4 \implies x = 2

Factoring out

2^x

from the terms allows us to isolate the exponential expression and find the value of

x

Key Concept

Sum of an Infinite Geometric Series and Exponential Properties

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Step-by-Step Solution

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