The first term of an arithmetic sequence is \frac{1}{2}, and the second term is \frac{5}{6… — ACT

The first term of an arithmetic sequence is $\frac{1}{2}$ , and the second term is $\frac{5}{6}$ . The first term of a geometric sequence is $2^2$ , and the common ratio is $2^3$ . Let $A$ be the third term of the arithmetic sequence, and let $G$ be the third term of the geometric sequence. If a third value, $V$ , is defined as $\frac{5}{6}$ less than twice $G$ , what is the value of $V + A$ ?

A
$\frac{1541}{3}$
B
$512$
$\frac{1537}{3}$ Answer
D
$-510$
E
$\frac{24577}{3}$

Answer

\frac{1537}{3}

The correct answer is

\frac{1537}{3}

because calculating the third term of the arithmetic sequence yields

A = \frac{7}{6}

and the third term of the geometric sequence yields

G = 256

. Translating the relationship for

V

yields

V = 2(256) - \frac{5}{6} = 512 - \frac{5}{6}

. Finding the sum of

V

and

A

yields

512 - \frac{5}{6} + \frac{7}{6} = 512\frac{1}{3} = \frac{1537}{3}

Step-by-Step Solution

Calculate the common difference

d

of the arithmetic sequence.

d = a_2 - a_1 = \frac{5}{6} - \frac{1}{2} = \frac{1}{3}

The common difference is the difference between any term and the preceding term in an arithmetic sequence.

Calculate the third term

A

of the arithmetic sequence.

A = a_1 + 2d = \frac{1}{2} + 2\left(\frac{1}{3}\right) = \frac{7}{6}

The

n

-th term of an arithmetic sequence is given by

a_n = a_1 + (n-1)d

Calculate the third term

G

of the geometric sequence.

G = g_1 \cdot r^2 = 2^2 \cdot (2^3)^2 = 2^2 \cdot 2^6 = 2^8 = 256

The

n

-th term of a geometric sequence is given by

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

Set up and solve for

V

using the algebraic relationship described.

V = 2G - \frac{5}{6} = 2(256) - \frac{5}{6} = 512 - \frac{5}{6}

The phrase '5/6 less than twice G' translates to

2G - \frac{5}{6}

Compute the sum of

V

and

A

V + A = \left(512 - \frac{5}{6}\right) + \frac{7}{6} = 512 + \frac{2}{6} = 512 + \frac{1}{3} = \frac{1537}{3}

Substitute the values of

V

and

A

and simplify the resulting fractional expression.

Key Concept

Arithmetic and Geometric Sequences and Series

Estimated Time:1m 30s

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