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 a1,a2,a3, is an arithmetic sequence with common difference d, show that the sequence 10a1,10a2,10a3, is a geometric sequence, and find the common ratio.
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Expert Answer

Raheem Donnelly
Answered 2021-08-06 Author has 75 answers
Given,
a1,a2,a3, is a n arithmetic sequence with common difference d.
Now, a2=a1+d,a3=a1+2d, ----- [an=a1+(n1)d].
Now consider the sequence, 10a1,10a2,10a3,
Here, 10a210a1=10a2+d10a1
=10d
10a310a2=10a1+2d10a1+d
=102d10d
=10d
Clearly,
second termfirst term=third termsecond term=10d
Therefore 10a1,10a2,10a3, is a geometric sequence.
Step 2
The common ratio is 10d.
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