Function as Average

Let xk = (-1)k for any positive integer k. Let f(n) = (x1 + x2 + … + xn)/n, where n is a positive integer. Give the range of this function.

  1. 0
  2. 1/n (where n is any positive integer)
  3. 0 and -1/n (where n is any odd positive integer)
  4. 0 and 1/n (where n is any positive integer)
  5. 1 and 1/n (where n is any odd positive integer).


Show/Hide Solution

xk = (-1)k
x1 = -1
x2 = 1
x3 = -1
x4 = 1
.
.
.
xn = -1 if n is odd
1 if n is even.

Thus x1 + x2 + … + xn = 0 if n is even, -1 if n is odd.

So

f(1) = -1/1 = -1
f(2) = 0/2 = 0
f(3) = -1/3
f(4) = 0/4 = 0
f(5) = -1/5
etc.

So the answer is (c).

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