Step 1

Given arithmetic sequence,

\(\displaystyle{7},{3},-{1},\ldots\ldots..,-{89}\)

n-th term formula is:

\(\displaystyle{a}_{{{n}}}={a}_{{{1}}}+{\left({n}-{1}\right)}{d}\)

Where,

\(\displaystyle{a}_{{{1}}}=\text{first term of the sequence}={7}\)

\(\displaystyle{a}_{{{n}}}=\text{last term of the sequence}=-{89}\)

\(\displaystyle{d}=\text{common difference}={\left({a}_{{{2}}}-{a}_{{{1}}}\right)}={\left({3}-{7}\right)}={4}\)

\(\displaystyle{n}=\text{number of terms}=?\)

Step 2

Substitute all values in the formula and solve for n,

\(\displaystyle-{89}={7}+{\left({n}-{1}\right)}{\left(-{4}\right)}\)

\(\displaystyle-{89}-{7}=-{4}{\left({n}-{1}\right)}\)

\(\displaystyle-{96}=-{4}{\left({n}-{1}\right)}\)

\(\displaystyle{\left({n}-{1}\right)}={\frac{{{\left(-{96}\right)}}}{{{\left(-{4}\right)}}}}\)

\(\displaystyle{\left({n}-{1}\right)}={24}\)

\(\displaystyle{n}={25}\)

Therefore, there are 25 terms in the sequence.

Given arithmetic sequence,

\(\displaystyle{7},{3},-{1},\ldots\ldots..,-{89}\)

n-th term formula is:

\(\displaystyle{a}_{{{n}}}={a}_{{{1}}}+{\left({n}-{1}\right)}{d}\)

Where,

\(\displaystyle{a}_{{{1}}}=\text{first term of the sequence}={7}\)

\(\displaystyle{a}_{{{n}}}=\text{last term of the sequence}=-{89}\)

\(\displaystyle{d}=\text{common difference}={\left({a}_{{{2}}}-{a}_{{{1}}}\right)}={\left({3}-{7}\right)}={4}\)

\(\displaystyle{n}=\text{number of terms}=?\)

Step 2

Substitute all values in the formula and solve for n,

\(\displaystyle-{89}={7}+{\left({n}-{1}\right)}{\left(-{4}\right)}\)

\(\displaystyle-{89}-{7}=-{4}{\left({n}-{1}\right)}\)

\(\displaystyle-{96}=-{4}{\left({n}-{1}\right)}\)

\(\displaystyle{\left({n}-{1}\right)}={\frac{{{\left(-{96}\right)}}}{{{\left(-{4}\right)}}}}\)

\(\displaystyle{\left({n}-{1}\right)}={24}\)

\(\displaystyle{n}={25}\)

Therefore, there are 25 terms in the sequence.