 Find a recurrence relation satisfied by this sequence. a_{n}=5^{n} jernplate8 2021-09-08 Answered
Find a recurrence relation satisfied by this sequence.
$$\displaystyle{a}_{{{n}}}={5}^{{{n}}}$$

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Given:
$$\displaystyle{a}_{{{n}}}={5}^{{{n}}}$$
Let us first determine the first term by replacing n in the given expression for $$\displaystyle{a}_{{{n}}}$$ by 0:
$$\displaystyle{a}_{{{0}}}={5}^{{{0}}}={1}$$
Let us similarly determine the next few terms as well:
$$\displaystyle{a}_{{{1}}}={5}^{{{1}}}={5}={5}{a}_{{{0}}}$$
$$\displaystyle{a}_{{{2}}}={5}^{{{2}}}={25}={5}{a}_{{{1}}}$$
$$\displaystyle{a}_{{{3}}}={5}^{{{3}}}={125}={5}{a}_{{{2}}}$$
We note that each term is the previous term multiplied by 5:
$$\displaystyle{a}_{{{n}}}={5}{a}_{{{n}-{1}}}$$
Thus a recurrence relation for $$\displaystyle{a}_{{{n}}}$$ is then:
$$\displaystyle{a}_{{{0}}}={1}$$
$$\displaystyle{a}_{{{n}}}={5}{a}_{{{n}-{1}}}$$
Note: There are infinitely many different recurence relations that satisfy any sequence.