For how many integer values of n in the interval -15 \le n \le 15 is the value of the expr… — GRE General Test

For how many integer values of $n$ in the interval $-15 \le n \le 15$ is the value of the expression $(-1)^{n^2 + n} - (-1)^{3n}$ equal to $2$ ?

Answer: 16

Answer

The correct answer is 16.

Because

n^2 + n = n(n + 1)

is the product of two consecutive integers, it is guaranteed to be even for every integer

n

. Consequently,

(-1)^{n^2 + n} = 1

. Substituting this into the given equation yields

1 - (-1)^{3n} = 2

, which reduces to

(-1)^{3n} = -1

. A power of

-1

equals

-1

if and only if the exponent is odd, so

3n

must be odd, which requires

n

itself to be odd. In the interval

[-15, 15]

, there are 16 odd integers: 8 negative odd integers and 8 positive odd integers.

Step-by-Step Solution

Determine the parity of

n^2 + n

The expression

n^2 + n = n(n + 1)

represents the product of two consecutive integers. Because one of any two consecutive integers is even, their product is always even. Therefore,

(-1)^{n^2 + n} = 1

for all integers

n

Simplifying the exponent with a known parity rule reduces the expression to a constant.

Isolate

(-1)^{3n}

in the equation

Substituting

1

into the original equation gives

1 - (-1)^{3n} = 2

, which simplifies to

(-1)^{3n} = -1

Isolating the exponential term reveals the sign condition required for the equality to hold.

Find the parity condition for

n

For

(-1)^{3n}

to equal

-1

, the exponent

3n

must be an odd integer. Since

3

is odd, the product

3n

is odd if and only if

n

is odd.

Applying the product parity rule (

\text{odd} \times \text{odd} = \text{odd}

) relates the condition on

3n

back to

n

Count the odd integers in the interval

[-15, 15]

The odd integers in the interval are

-15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15

. There are 16 such integers.

Counting all qualifying values within the specified range yields the final numeric answer.

Key Concept

Parity rules for consecutive integers and exponents of negative numbers

Question

Answer

Step-by-Step Solution

Key Concept

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