Mathematical Induction and Power Summations

S5 (n) = (1/ 12) [n (n + 1)]^2 [2 n^2 + 2 n - 1]

S5 (21) = 16417401 = 1.6417401 x 10^7 --> 1.642 x 10^7

In the proof by mathematical induction, one must prove that the equation works for the base value (i.e. $n=1$ ) and that it also works from one arbitrary value of $n$ to the next.

$\sum _{ k=1 }^{ 1 }{ { k }^{ 5 } }$ = $1^5$ = 1

and if you plug the value of $n=1$ into each of these three equations, you'll see that they all yield '1', therefore all pass the base step.

It would take too much time to do the inductive step on all three equations, but if you did, it would be noticed that only the final equation is the correct representation of the original summation. Here is the inductive step for equation 3:

Assuming equation is true for $n=p$ :

$\sum_{k=1}^{p}{{k}^{5}} = \frac{p^2}{12}(2p^4+6p^3+5p^2-1)$ AND

$\sum_{k=1}^{p+1}{{k}^{5}}=\frac{(p+1)^2}{12}(2(p+1)^4+6(p+1)^3+5(p+1)^2-1)$

Following the progression: $\sum_{k=1}^{p+1}{{k}^{5}}=\sum_{k=1}^{p}{{k}^{5}}+(p+1)^5$

So if we can prove that from one value, $(p$ , we can get to the next value, $p+1$ , using the equation: we have proved the inductive step, and thus the equation. Therefore we must prove that:

$\sum_{k=1}^{p}{{k}^{5}}+(p+1)^5$ = $\frac{(p+1)^2}{12}(2(p+1)^4+6(p+1)^3+5(p+1)^2-1)$

$\Rightarrow \frac{p^2}{12}(2p^4+6p^3+5p^2-1)+(p+1)^5=\frac{(p+1)^2}{12}(2(p+1)^4+6(p+1)^3+5(p+1)^2-1)$

My natural instincts just drive me to want to expand everything I see, so there are perhaps faster ways to view both sides equally, however, I like to expand. Expanding the left hand side (LHS) first:

$\Rightarrow \frac{1}{12}(2p^6+18p^5+65p^4+120p^3+119p^2+60p^2+12)$

Now, the right hand side (RHS): $\Rightarrow \frac{1}{12}(2(p+1)^6+6(p+1)^5+5(p+1)^4-(p+1)^2)$ $\Rightarrow \frac{1}{12}(2p^6+18p^5+65p^4+120p^3+119p^2+60p^2+12)$

$\therefore LHS=RHS \Box$ . This equation is now proven.

Now for STEP 2 , which is to simply substitute 21 into the original equation:

$\frac{21^2}{12}(2(21)^4+6(21)^3+5(21)^2-1) = 16,417,401$

which can be represented as $1.6417401\times 10^7$ or, to 4 significant figures:

$\boxed {1.642}$