Subject: induction
Name: Zamira
Who are you: Student

i have a problem with this mathematical induction:
(1^5)+(2^5)+(3^5)+...+(n^5) = ((n^2)*((n+1)^2)*((2n^2)+2n-1))/12

thanks for your help and time

zamira

Hi Zamira,

For the inductive step assume that the statement is true when n = k - 1, that is

1⁵ + 2⁵ + 3⁵ + ... + (k-1)⁵ = (k-1)² k² (2(k-1)² +2(k-1) - 1)/12

This simplifies to

1⁵ + 2⁵ + 3⁵ + ... + (k-1)⁵ = (k-1)² k² (2k² - 2k - 1)/12       (*)

The task now is to use equation * to show that

1⁵ + 2⁵ + 3⁵ + ... + k⁵ = k² [(k+1)² (2k² + 2k - 1)]/12       (**)

From equation *

1⁵ + 2⁵ + 3⁵ + ... + (k-1)⁵ + k⁵
= (k-1)² k² (2k² - 2k - 1)/12 + k⁵
= k²/12 [(k-1)² (2k² - 2k - 1) + 12k³]       (***)

Expand the expressions inside the square brackets in equations ** and *** and cpm pare.

Penny