The value of 1^3-2^3+3^3-+15^3 is:

Question

The value of $1^3-2^3+3^3-\ldots+15^3$ is:

Accepted Answer

We need to find

${1}^{3}-{2}^{3}+{3}^{3}-{4}^{3}+\cdots +{15}^{3}$

This is an alternating sum of cubes.

Let us group the terms in pairs:

$({1}^{3}-{2}^{3})+({3}^{3}-{4}^{3})+({5}^{3}-{6}^{3})+\cdots +({13}^{3}-{14}^{3})+{15}^{3}$

Now calculate each pair:

${1}^{3}-{2}^{3}=1-8=-7$

${3}^{3}-{4}^{3}=27-64=-37$

${5}^{3}-{6}^{3}=125-216=-91$

${7}^{3}-{8}^{3}=343-512=-169$

${9}^{3}-{10}^{3}=729-1000=-271$

${11}^{3}-{12}^{3}=1331-1728=-397$

${13}^{3}-{14}^{3}=2197-2744=-547$

Also,

${15}^{3}=3375$

Now add all these:

$-7-37-91-169-271-397-547+3375$

First add the negative terms:

$7+37+91+169+271+397+547=1519$

So,

$3375-1519=1856$

Hence, the value is

$1856$

So, the correct option is B.

Solutions · 1