# I need help with an algebra problem

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### Question Description

The sequences below are either arithmetic or geometric.  For each sequence, determine whether i it is arithmetic or geometric, and write the formula for the nth term an of that sequence

TYPE               nth term formula

16, 19, 22, ...

7, 21, 63, ...

shyamnair3
School: UIUC

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16, 19, 22, ...

$\\ a_1=16\\ \\ a_2=19\\ \\ a_3=22\\ \\ a_2-a_1=19-16=3\\ \\ a_3-a_2=22-19=3\\ \\ \therefore a_3-a_2=a_2-a_1=3$

So the sequence has a common difference of 3 and hence is an arithmetic sequence.

The nth term of the sequence is

$\\ a_n=a_1+(n-1)d\\ \\ a_n=16+(n-1)3\\ \\ a_n=16+3n-3\\ \\ a_n=13+3n\\$

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7, 21, 63, ...

$\\ a_1=7\\ \\ a_2=21\\ \\ a_3=63\\ \\ \frac{a_2}{a_1}=\frac{21}{7}=3\\ \\ \frac{a_3}{a_2}=\frac{63}{21}=3\\ \\ \therefore \frac{a_3}{a_2}=\frac{a_2}{a_1}=3\\$

So the sequence has a common ratio of 3 and hence is a geometric sequence.

The nth term of the sequence is

$\\ a_n=a_1r^{n-1}\\ \\ a_n=7\times3^{n-1}\\$

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Anonymous
awesome work thanks

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