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sum of fifth powers of first n natural numbers

Posted October 12th, 2008 by SD_88
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how do you find the sum of first n natural no.s..............can anyone please tell me directly also what is the sum of first n natural no.s.........i.e $ 1^5 +2^5 + 3^5 + 4^5.............+ n^5= $????

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hi

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answer

On October 12th, 2008 Structure says:

Let

$$S_i=\sum_{k=0}^{k=n}k^i$$

You want to find

$$S_5=\sum_{k=0}^{k=n}k^5$$

It is very easy to find in out with a little work
You start with

$$(k+1)^6=k^6+6k^5+15k^4+20k^3+15k^2+6k+1$$

then sum up for k from 1 to n
Reducing terms you have

$$(n+1)^6=1+6S_5+15S_4+20S_3+15S_2+6S_1+n$$

From this you find $ S_5 $ if you know $ S_4 $, $ S_3=\frac{n^2(n+1)^2}{4} $ $ S_2=\frac{n(n+1)(2n+1)}{6} $ and $ S_1=\frac{n(n+1)}{2} $
Try to find yourself $ S_4 $ from

$$(n+1)^5=1+5S_4+10S_3+10S_2+5S_1+n$$

sum-fifth-powers-first-n-natural

On October 12th, 2008 SD_88 says:
$$(k+1)^6=k^6+6k^5+15k^4+20k^3+15k^2+6k+1$$

then you summed up for k from 1 to n
and you Reduced certain terms
and you got

$$(n+1)^6=1+6S_5+15S_4+20S_3+15S_2+6S_1+n$$

how did you get

$$(n+1)^6$$

on lhs...and summation of $ k^6 $ on rhs as 1..cud you give steps...rest all the terms are ok on rhs...shouldnt we get an S6 on rhs from the first term....
i would be thankful to you if you cud give the finl result too if possible.....or a source where it wud be present.....as i am more interested in the reswult rather than its derivation and method of getting it.....the result of summation of $ k^5 $ is of utmost importance to me now...
THANK YOU for answering me!!

Sum of powers of 3 & 5's

On December 14th, 2009 RALPHINEVERETT says:

Sum of Cubes:
1^3 + 2^3 +3^3 +4^3-----------N^3 = sum to N squared. ( (N^2 +N)/2)^2
A= sum 1 to N . Example : The sum to 4 = 1 +2+3+4= 10 = 10^2 = 100
1 + 8 + 27 + 64 = 100 =A^2

Sum of N^5

A= sum 1 to N

Sum of N^5 = (4A^3 - A^2)/3

Example: N = 3 A = 6

1^5 +2^5 +3^5 = 1 + 32 + 243 = 276

(4X6^3 - 6^2)/3 = 276

Example: N=4 A=10

(4(10)^3 - 10^2)/3 = 4000 - 100 = 3900 3900/3 = 1300

1^5 + 2^5 + 3^5 + 4^5 = 1+ 32 + 243 + 1024 = 1300

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