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 x k2 x (2(k-1)2 +2(k-1) - 1)/12

This simplifies to

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

The task now is to use equation * to show that

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

From equation *

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

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

Penny