let-U-n-n-2-if-n-even-and-U-n-n-1-2-if-n-odd-let-f-n-k-0-n-U-k-prove-that-x-y-N-2-f-x-y-f-x-y-xy-

Question Number 73039 by mathmax by abdo last updated on 05/Nov/19

l e t U_{n} = \frac{n}{2} i f n e v e n a n d U_{n} = \frac{n - 1}{2} i f n o d d l e t f (n) = \sum_{k = 0}^{n} U_{k}

p r o v e t h a t \forall (x, y) \in N^{2} f (x + y) - f (x - y) = x y

Answered by mind is power last updated on 05/Nov/19

just for the definition of f

x ⩾ y

x + y, x - y has sam parite

f (n) = \sum_{k = 0}^{n} U_{k} = \underset{k = 0}{\sum^{E (\frac{n}{2})}} k + \underset{k = 1}{\sum^{E (\frac{n - 1}{2})}} k = \frac{(E (\frac{n}{2}) + 1) . (E (\frac{n}{2}))}{2} + \frac{E (\frac{n - 1}{2}) . (1 + E (\frac{n - 1}{2}))}{2}

f (x + y) - f (x - y)

is x + y = 2 n

x - y = 2 n - 2 y

\Rightarrow f (x + y) - f (x - y) = \frac{n (n + 1)}{2} - \frac{(n - y + 1) (n - y)}{2} + \frac{(n - 1) (n)}{2} - \frac{(n - y - 1) (n - y)}{2}

= \frac{n^{2} + n - (n^{2} - 2 ny + y^{2} + n - y) + n^{2} - n - (n^{2} + y^{2} - 2 ny - n + y)}{2}

= \frac{- {2 y}^{2} + 4 ny}{2} = \frac{- {2 y}^{2} + 2 (x + y) y}{2} = xy

same thing if x + y = 2 n + 1

Commented by mathmax by abdo last updated on 05/Nov/19

t h a n x s i r .