If k is a positive integer such that \frac{3^{k+2} + 3^{k+4}}{10} = 3^{2k-1}, what is the … — GMAT

If $k$ is a positive integer such that $\frac{3^{k+2} + 3^{k+4}}{10} = 3^{2k-1}$ , what is the value of $k$ ?

The value of

k

is 3.

Factoring

3^{k+2}

from the numerator gives

3^{k+2}(1 + 3^2) = 3^{k+2}(10)

. Dividing by 10 simplifies the left side to

3^{k+2}

. Setting

3^{k+2} = 3^{2k-1}

requires

k + 2 = 2k - 1

, which yields

k = 3

Factor out the common exponential term

3^{k+2}

from the numerator on the left-hand side.

3^{k+2} + 3^{k+4} = 3^{k+2}(1 + 3^2) = 3^{k+2}(1 + 9) = 10 \cdot 3^{k+2}

When adding terms with identical bases, factor out the term with the smallest exponent to simplify the sum.

Substitute the factored expression into the fraction and simplify.

\frac{10 \cdot 3^{k+2}}{10} = 3^{k+2}

Canceling the common factor of 10 in the numerator and denominator simplifies the left-hand side.

Set the simplified left-hand side equal to the right-hand side of the original equation.

3^{k+2} = 3^{2k-1}

Both sides now have the same base of 3.

Equate the exponents since the bases are equal and solve for

k

k + 2 = 2k - 1 \implies 2 + 1 = 2k - k \implies k = 3

For any non-zero, non-one base

b

b^x = b^y

implies

x = y

Factoring exponential expressions with identical bases and equating exponents.

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