A sequence is defined by the formula a_n = 2 + 3 \cdot \frac{2^n \cdot 2^{n-1}}{6} for all… — ACT

A sequence is defined by the formula $a_n = 2 + 3 \cdot \frac{2^n \cdot 2^{n-1}}{6}$ for all integers $n \geq 1$ . What is the value of the second term, $a_2$ ?

A
$4$
$6$ Answer
C
$\frac{20}{3}$
D
$\frac{26}{7}$
E
$10$

Answer

The value of the second term is 6.

The correct answer is 6. When substituting

n = 2

into the formula, the numerator simplifies to

2^2 \cdot 2^1 = 8

. The fraction becomes

\frac{8}{6} = \frac{4}{3}

. Multiplying by 3 yields 4, and adding 2 yields 6.

Step-by-Step Solution

Substitute

n = 2

into the sequence formula.

a_2 = 2 + 3 \cdot \frac{2^2 \cdot 2^{2-1}}{6}

To find the second term, we replace

n

with 2 throughout the expression.

Simplify the exponents in the numerator.

2^2 \cdot 2^{2-1} = 4 \cdot 2^1 = 4 \cdot 2 = 8

Evaluate the exponential terms in the numerator before performing other operations.

Substitute the simplified numerator back and evaluate the fraction.

\frac{8}{6} = \frac{4}{3}

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 2.

Multiply by 3 and add 2 to get the final answer.

2 + 3 \cdot \frac{4}{3} = 2 + 4 = 6

Perform multiplication before addition according to the order of operations.

Key Concept

Evaluating terms of a sequence given by an explicit formula involving exponent rules.

Estimated Time:1m 0s

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