Prove-the-Identity-for-any-a-n-in-Real-Number-1-a-a-n-a-a-2-n-2-a-2-n-1-2-Greatest-Integer-Function-

Question Number 162414 by HongKing last updated on 29/Dec/21

Prove the Identity for any (a, n) in Real Number

(1 + a) \cdot a^{[n]} = a \cdot a^{2 [\frac{n}{2}]} + a^{2 [\frac{n + 1}{2}]}

[*] Greatest Integer Function

Answered by mindispower last updated on 31/Dec/21

n \in [2 k, 2 k + 1 [

[n] = 2 k

(1 + a) . a^{2 k} = a . a^{2 k} + a^{2 k} = (1 + a) . a^{2 k}

t r u e

n \in [2 k - 1, 2 k]

[n] = 2 k - 1

[\frac{n}{2}] = k - 1

[\frac{n + 1}{2}] = k

(1 + a) . a^{2 k - 1} = ? a . a^{2 (k - 1)} + a^{2 k} = a^{2 k - 1} + a^{2 k} = a^{2 k - 1} (1 + a)

T r u e

(1 + a) . a^{[n]} = a . a^{2 [\frac{n}{2}]} + a^{2 [\frac{n + 1}{2}]}

Commented by HongKing last updated on 31/Dec/21

cool my dear Sir thank you so much