Question

derive the formula for sigma n power 4

Answer 1

Heya

Using Binomial Expansion :
${x}^{5} - {(x - 1)}^{5} = 5 {x}^{4} - 10 {x}^{3} + 10 {x}^{2} - 5x + 1$

Summing the R.H.S. and L.H.S. from n to 1 :

${n}^{5} = 5 \sum {n}^{4} - 10\sum {n}^{ 3} + 10\sum {n}^{2} - 5\sum {n} + n$

Putting the formula for the sums we know about :

$\sum {n}^{3} = {( \frac{n(n + 1)}{2} )}^{2}$

$\sum {n}^{2} = \frac{n(n + 1)(2n + 1)}{6}$

$\sum {n} = \frac{n(n + 1)}{2}$


Putting these formula in above relation and simplifying :

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