evalute-the-value-of-m-2-5-m-4-

Question Number 9279 by j.masanja06@gmail.com last updated on 28/Nov/16

evalute the value of

{\sum^{5}}_{m = 2} m^{4}

Answered by mrW last updated on 28/Nov/16

please refer to Q 9232 and Q 9265 .

Answered by geovane10math last updated on 28/Nov/16

2^{4} + 3^{4} + 4^{4} + 5^{4} + 6^{4} =

2^{4} = 16

3^{4} = 81

4^{4} = 256

5^{4} = 625

6^{4} = 1296

16 + 81 + 256 + 625 + 1296 =

= 97 + 881 + 1296 = 1296 + 978 =

2274

Answered by mrW last updated on 28/Nov/16

generally for 1 ⩽ m ⩽ n

\underset{k = 1}{\sum^{n}} k^{4} = \frac{n (n + 1) (2 n + 1) ({3 n}^{2} + 3 n - 1)}{30}

\underset{k = 1}{\sum^{m - 1}} k^{4} = \frac{(m - 1) m (2 m - 1) ({3 m}^{2} - 3 m - 1)}{30}

\underset{k = m}{\sum^{n}} k^{4} = \underset{k = 1}{\sum^{n}} k^{4} - \underset{k = 1}{\sum^{m - 1}} k^{4}

\underset{k = m}{\sum^{n}} k^{4} = \frac{n (n + 1) (2 n + 1) ({3 n}^{2} + 3 n - 1) - (m - 1) m (2 m - 1) ({3 m}^{2} - 3 m - 1)}{30}

\underset{k = 2}{\sum^{5}} k^{4} = \frac{5 \times 6 \times 11 \times (3 \times 5^{2} + 3 \times 5 - 1) - 1 \times 2 \times 3 \times (3 \times 2^{2} - 3 \times 2 - 1)}{30}

= \frac{5 \times 6 \times 11 \times 89 - 6 \times 5}{30} = 978

or

2^{4} + 3^{4} + 4^{4} + 5^{4} = 16 + 81 + 256 + 625 = 978

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