"Are there any generalisations of the identity ∑k=1nk3=(∑k=1nk)2..." was answered by our community

ransomedv37

Answered question

2022-03-15

Are there any generalisations of the identity

\sum_{k = 1}^{n} k^{3} = {(\sum_{k = 1}^{n} k)}^{2}

Ettrapinithr

Beginner2022-03-16Added 7 answers

We can't have a relationship of the form

\forall n \in N^{\cdot}, \sum_{k = 1}^{n} k^{a} = {(\sum_{k = 1}^{n} k^{b})}^{c}

for

a, b, c \in N

, except in the case

c = 1

and

a = b

a = 3, b = 1

and

c = 2

. Indeed, we can write

\sum_{k = 1}^{n} k^{a} = n^{a + 1} \frac{1}{n} \sum_{k = 1}^{n} {(\frac{k}{n})}^{a}

and if we have the initial equality we should have

a + 1 = (b + 1) c

and

a + 1 = {(b + 1)}^{c}

. In particular,

{(b + 1)}^{c - 1} = c

. If c>1, then

c = {(b + 1)}^{c - 1} \geq 2^{c - 1} \geq c

, and we should have c=2 and b=1 therefore a=3

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