Question

What is the nth term?

Answer 1

Let
`a_k` denote the kth term of the sequence. Then

`a_k=a_1+d(k-1)`

where d is the common difference between consecutive terms in the sequence and a₁ is the first term.

The sum of the first n terms is

`S_n=\displaystyle\sum_(k=1)^na_k=a_1+a_2+\cdots+a_(n-1)+a_n`

From the formula for
`a_k`, we get

`S_n=\displaystyle\sum_(k=1)^n(a_1+d(k-1))=a_1\sum_(k=1)^n1+d\sum_(k=1)^n(k-1)`

`S_n=\displaystyle na_1+d\sum_(k=0)^(n-1)k`

`S_n=na_1+\frac{d(n-1)n}2`

`S_n=\frac n2(2a_1-d+dn)`

So we have
`d=-5`, and
`2a_1-d=16` so that
`a_1=\frac{11}2`.

Then the nth term in the sequence is

`a_n=\frac{11}2-5(n-1)=\boxed{\frac{21-10n}2}`