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For all integers $x>0$ , let $f(x)$ be defined as $f(x)=\frac{f(x-1)}{(-1)^x}$ . If $f(1)=1$ , which of the following statement is correct for the values of $f(x)$ ?

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Evaluate an exponential function

Full solution

Q. For all integers

x>0

, let

f(x)

be defined as

f(x)=\frac{f(x-1)}{(-1)^x}

. If

f(1)=1

, which of the following statement is correct for the values of

f(x)

?

Understand function and initial condition: Understand the function and initial condition. $\newline$ Given $f(x) = \frac{f(x-1)}{(-1)^x}$ and $f(1) = 1$ . $\newline$ We need to find the pattern of $f(x)$ for $x > 1$ .
Calculate $f(2)$ : Calculate $f(2)$ . $f(2) = \frac{f(1)}{(-1)^2} = \frac{1}{1} = 1$
Calculate $f(3)$ : Calculate $f(3)$ . $f(3) = \frac{f(2)}{(-1)^3} = \frac{1}{-1} = -1$
Calculate $f(4)$ : Calculate $f(4)$ . $f(4) = \frac{f(3)}{(-1)^4}$ $= \frac{-1}{1}$ $= -1$

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