Let x_{k} = (-1)^{k} for any positive integer k. Let f(n) = (x_{1} + x_{2} + … + x_{n})/n, where n is a positive integer. Give the range of this function.
- 0
- 1/n (where n is any positive integer)
- 0 and -1/n (where n is any odd positive integer)
- 0 and 1/n (where n is any positive integer)
- 1 and 1/n (where n is any odd positive integer).
x_{k} = (-1)^{k} x_{1} = -1 x_{2} = 1 x_{3} = -1 x_{4} = 1 . . . x_{n} = -1 if n is odd 1 if n is even. |
Thus x_{1} + x_{2} + … + x_{n} = 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.
f(2) = 0/2 = 0
f(3) = -1/3
f(4) = 0/4 = 0
f(5) = -1/5
etc.
So the answer is (c).