Find specific terms of an arithmetic sequence given other terms. How many terms are there in the following finite arithmetic sequence? 7,3,-1,\cdots,-89

allhvasstH 2021-08-08 Answered
Find specific terms of an arithmetic sequence given other terms.
How many terms are there in the following finite arithmetic sequence?
\(\displaystyle{7},{3},-{1},\cdots,-{89}\)
1) 23
2) 24
3) 25
4) 28
5) 30

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Expert Answer

Malena
Answered 2021-08-09 Author has 17718 answers
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.
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