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Find the common ratio

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    Vu Hung
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Problem Statement

In a geometric sequence, a2=12a_2=12 and a5=324a_5=324. Find the common ratio rr and the first term a1a_1.

Solutions

From an=arn−1a_n=ar^{n-1},

a2=ar=12,a5=ar4=324.a_2=ar=12,\qquad a_5=ar^4=324.

Divide to remove aa:

a5a2=r3=32412=27  ⟹  r=3.\frac{a_5}{a_2}=r^3=\frac{324}{12}=27 \implies r=3.

Then a1=a=123=4a_1=a=\frac{12}{3}=4.

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