(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?

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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,…..arnFor this sequence to be increasing:a≤ar≤ar2≤ar3≤...≤arn−1≤arnNow a≤araa≤r⇒1≤rThus, 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,…arnto be decreasing:a≥ar≥ar2≥ar3≥…≥arn−1≥arna≥ar⇒aa≥r⇒1≥rThus, 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 increasingNote that a general arithmetic sequence with common difference d and initial value a is represented as:a,a+d,a+2d,a+3d,…,a+ndTo be increasing:a≤a+d≤a+2d≤a+3d≤…≤a+(n−1)d≤a+nda≤a+d⇒a−a≤d⇒0≤dThus, 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.

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

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