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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Expert Answer

krolaniaN
Answered 2021-09-09 Author has 21953 answers
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.
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