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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