The nth term of an arithmetic sequence is \(\displaystyle{a}_{{{n}}}={a}_{{{1}}}+{\left({n}-{1}\right)}{d}\ {w}{h}{e}{r}{e}\ {a}_{{{1}}}\) is the first term and d is the common difference.

For the arithmetic sequence 15, 20, 25, 30, ... the first term is \(\displaystyle{a}_{{{1}}}={15}\) and the common difference is \(\displaystyle{d}={a}_{{{2}}}-{a}_{{{1}}}={20}—{15}={5}\). The nth term is then

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

\(\displaystyle{a}_{{{n}}}={15}+{\left({n}-{1}\right)}{\left({5}\right)}\) Substitute.

\(\displaystyle{a}_{{{n}}}={15}+{5}{n}-{5}\) Distribute.

\(\displaystyle{a}_{{{n}}}={10}+{5}{n}\) Combine like terms.

For the arithmetic sequence 15, 20, 25, 30, ... the first term is \(\displaystyle{a}_{{{1}}}={15}\) and the common difference is \(\displaystyle{d}={a}_{{{2}}}-{a}_{{{1}}}={20}—{15}={5}\). The nth term is then

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

\(\displaystyle{a}_{{{n}}}={15}+{\left({n}-{1}\right)}{\left({5}\right)}\) Substitute.

\(\displaystyle{a}_{{{n}}}={15}+{5}{n}-{5}\) Distribute.

\(\displaystyle{a}_{{{n}}}={10}+{5}{n}\) Combine like terms.