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What will be the sum of: 1^1 + 2^2 + 3^3 + 4^4.........+ nⁿ

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rnk | Student, Grade 10 | Honors

Posted July 7, 2011 at 7:11 PM via web

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What will be the sum of: 1^1 + 2^2 + 3^3 + 4^4.........+ nⁿ

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giorgiana1976 | College Teacher | Valedictorian

Posted July 7, 2011 at 11:31 PM (Answer #1)

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You could consider the sum ofthe powers as the followings:

1 + 2^2 + 3^3 + ... + n^n

1 + 2*2 + 3*3*3 + ..... + n*n*...*n

1 + 2 + 3 + ...... + n = Sn

2 + 3 + ...... + n = S(n-1)

3 + ....... + n = S(n-2)

..............................................

n - 1 + n = S1

n = S0

But we know that Sn = (1+n)*n/2, therefore S(n-1) = (1+n-1)*(n-1)/2 = n*(n-1)/2

The sum will become:

1 + 2^2 + 3^3 + ... + n^n = Sn + S(n-1) + ... + S1 + S0

1 + 2^2 + 3^3 + ... + n^n = (1+n)*n/2 + n*(n-1)/2 + ... + 0

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