(a) What are the conditions on the common ratio r and initial value a, that would make the resulting

Carl Hood

Carl Hood

Answered question

2022-03-03

(a) What are the conditions on the common ratio r and initial value a, that would make the resulting geometric sequence increasing?
(b) What are the conditions on the common ratio r and initial value a, that would make the resulting geometric sequence decreasing?
(c) What are the conditions on the common difference d and initial value a, that would make the resulting arithmetic sequence increasing?

Answer & Explanation

tardanetkd2

tardanetkd2

Beginner2022-03-04Added 9 answers

a) It is required to explain the condition on common ratio r and initial value a to make the resulting geometric sequence increasing.
General geometric sequence with common ratio r and initial value a is represented as:
a,ar,ar2,ar3,..arn
For this sequence to be increasing:
aarar2ar3...arn1arn
Now aar
aar1r
Thus, for geometric sequence to be increasing the condition on common ratio r should be greater than or equals to 1, and no condition om initial value a is required.
b) It is required to explain the condition on common ratio r and initial value a to make the resulting geometric sequence is decreasing.
a,ar,ar2,ar3,arn
to be decreasing:
aarar2ar3arn1arn
aar
aar1r
Thus, for geometric sequence to be decreasing the condition on common ratio r that it should be less than or equals to 1, and no condition on initial value a is required.
c) It is required to explain the condition on the common difference d and initial value a, that would make the resulting arithmetic sequence increasing
Note that a general arithmetic sequence with common difference d and initial value a is represented as:
a,a+d,a+2d,a+3d,,a+nd
To be increasing:
aa+da+2da+3da+(n1)da+nd
aa+daad0d
Thus, for arithmetic sequence to be increasing the condition on common difference d that it should be greater than or equals to 0, and no condition on initial value a is required.

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