 # 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. 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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Given,
${a}_{1},{a}_{2},{a}_{3},\cdots$ is a n arithmetic sequence with common difference d.
Now, ${a}_{2}={a}_{1}+d,{a}_{3}={a}_{1}+2d$, ----- $\left[\therefore {a}_{n}={a}_{1}+\left(n-1\right)d\right]$.
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 ${10}^{d}$.