Question

Give the first six terms of the following sequences. You can assume that the seq

Sequences
ANSWERED
asked 2021-09-13
Give the first six terms of the following sequences. You can assume that the sequences start with an index of 1.
1) An arithmetic sequence in which the first value is 2 and the common difference is 3.
2) A geometric sequence in which the first value is 27 and the common ratio is \(\displaystyle{\frac{{{1}}}{{{3}}}}\)

Expert Answers (1)

2021-09-14
1)
The value \(\displaystyle{u}_{{1}}={2}\)
Common difference= 3
use the formula
\(\displaystyle{u}_{{n}}={u}_{{1}}+{\left({n}-{1}\right)}{d}\)
Replace \(\displaystyle{u}_{{1}}={2},\ {d}={3}\)
\(\displaystyle\Rightarrow{u}_{{n}}={2}+{\left({n}+{1}\right)}{3}\)
\(\displaystyle\Rightarrow{3}{n}-{3}+{2}\)
\(\displaystyle\Rightarrow{u}_{{n}}={3}{n}-{1}\) \(\displaystyle{n}={1}\ {u}_{{1}}={2}\)
\(\displaystyle{n}={2}\ {u}_{{2}}={5}\)
\(\displaystyle{n}={3}\ {u}_{{3}}={8}\)
\(\displaystyle{n}={4}\ {u}_{{4}}={11}\)
\(\displaystyle{n}={5}\ {u}_{{5}}={14}\)
\(\displaystyle{n}={6}\ {u}_{{6}}={17}\)
2)
a= first term= 27
\(\displaystyle\gamma\)= common ration= \(\displaystyle{\frac{{{1}}}{{{3}}}}\)
We know the formula
\(\displaystyle{a}_{{n}}={a}\gamma^{{{n}-{1}}}\)
Replace a=27, \(\displaystyle\gamma={\frac{{{1}}}{{{3}}}}\)
\(\displaystyle{a}_{{n}}={27}{\left({\frac{{{1}}}{{{3}}}}\right)}^{{{n}-{1}}}\)
\(\displaystyle{a}_{{n}}={\left({3}\right)}^{{{3}}}{\left({\frac{{{1}}}{{{3}}}}\right)}^{{{n}-{1}}}\) \(\displaystyle{a}_{{n}}={3}^{{{3}}}\cdot{3}^{{-{\left({n}-{1}\right)}}}\)
\(\displaystyle{a}_{{n}}={3}^{{{4}-{n}}}\) \(\displaystyle{a}_{{1}}={27},\ {a}_{{2}}={9},\ {a}_{{3}}={3},\ {a}_{{4}}={0},\ {a}_{{5}}={\frac{{{1}}}{{{3}}}},{a}_{{6}}={\frac{{{1}}}{{{9}}}}\)
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