(a) State the standard closed‐form expressions for each of the following finite sums…

MATHEMATICS · Subjective · ISC · Mathematics · Algebra · Algebra

Problem

(a) State the standard closed‐form expressions for each of the following finite sums of natural numbers up to

n

: (i)

\displaystyle\sum_{k=1}^{n} k

(ii)

\displaystyle\sum_{k=1}^{n} k^{2}

(iii)

\displaystyle\sum_{k=1}^{n} k^{3}

(b) Prove that for every positive integer

n

, the identity

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

holds true. (c) Using the result obtained in part (b), find the exact value of

\displaystyle\sum_{k=1}^{30} k^{3}