Hi Zamira,
For the inductive step assume that the statement is true when n = k  1, that is
1^{5} + 2^{5} + 3^{5} + ... + (k1)^{5} = (k1)^{2} k^{2} (2(k1)^{2} +2(k1)  1)/12
This simplifies to
1^{5} + 2^{5} + 3^{5} + ... + (k1)^{5} = (k1)^{2} k^{2} (2k^{2}  2k  1)/12 (*)
The task now is to use equation * to show that
1^{5} + 2^{5} + 3^{5} + ... + k^{5} = k^{2} [(k+1)^{2} (2k^{2} + 2k  1)]/12 (**)
From equation *
1^{5} + 2^{5} + 3^{5} + ... + (k1)^{5} + k^{5}
= (k1)^{2} k^{2} (2k^{2}  2k  1)/12 + k^{5}
= k^{2}/12 [(k1)^{2} (2k^{2}  2k  1) + 12k^{3}] (***)
Expand the expressions inside the square brackets in equations ** and *** and cpm pare.
Penny
