# Mathematical Induction and Power Summations

Algebra Level pending

STEP 1

The series $$\sum _{ k=1 }^{ n }{ { k }^{ 5 } }$$ can be evaluated with one of the following formulae:

1) $$2n^2(n^4-n^3+\frac{1}{2}n^2)$$

2)$$\frac{n}{10}(4n^5+5n^2+2n-1$$)

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

Use mathematical induction to prove which formula can be used to evaluate the summation for any positive integer value of $$n$$.

STEP 2

The series $$\sum _{ k=1 }^{ 21 }{ { k }^{ 5 } }$$ can be expressed as $$q\times 10^7$$.

Find $$q$$ to 4 significant figures.

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