kuCAu
2021-08-05
Answered

Exponentials of an Arithmetic Sequence, If ${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence with common difference d, show that the sequence ${10}^{{a}_{1}},{10}^{{a}_{2}},{10}^{{a}_{3}},\cdots$ is a geometric sequence, and find the common ratio.

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Raheem Donnelly

Answered 2021-08-06
Author has **75** answers

Given,

${a}_{1},{a}_{2},{a}_{3},\cdots$ is a n arithmetic sequence with common difference d.

Now,${a}_{2}={a}_{1}+d,{a}_{3}={a}_{1}+2d$ , ----- $[\therefore {a}_{n}={a}_{1}+(n-1)d]$ .

Now consider the sequence,${10}^{{a}_{1}},{10}^{{a}_{2}},{10}^{{a}_{3}},\cdots$

Here,$\frac{{10}^{{a}_{2}}}{{10}^{{a}_{1}}}=\frac{{10}^{{a}_{2}^{+d}}}{{10}^{{a}_{1}}}$

$={10}^{d}$

$\frac{{10}^{{a}_{3}}}{{10}^{{a}_{2}}}=\frac{{10}^{{a}_{1}^{+2d}}}{{10}^{{a}_{1}^{+d}}}$

$=\frac{{10}^{2d}}{{10}^{d}}$

$={10}^{d}$

Clearly,

$\frac{\text{second term}}{\text{first term}}=\frac{\text{third term}}{\text{second term}}={10}^{d}$

Therefore${10}^{{a}_{1}},{10}^{{a}_{2}},{10}^{{a}_{3}},\cdots$ is a geometric sequence.

Step 2

The common ratio is$10}^{d$ .

Now,

Now consider the sequence,

Here,

Clearly,

Therefore

Step 2

The common ratio is

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