Question

Use Gauss's approach to find the following sum

4+10+16+22+...+70

The sum of the sequence is

Answer 1

Each consecutive term in the sum is separated by a difference of 6, so the
`n`-th term is
`4+6(n-1)=6n-2` for
`n\ge1`. The last term is 70, so there are
`6n-2=70\implies n=12` terms in the sum.

Now,

`S=4+10+\cdots+64+70`

but also

`S=70+64+\cdots+10+4`

Doubling the sum and grouping terms in the same position gives

`2S=(4+70)+(10+64)+\cdots+(64+10)+(70+4)=12\cdot74`

`\implies\boxed{S=444}`