Answer to Question #6819 in Real Analysis for Akhtar Rasul

Question

Dear Expert sir i have anohter question.give me an example of a sequence which is both geometric as well as arithmetic.a lot of thanks .

Answer 1

The formula for an arithmetic sequence is: a(n)=a(1)+d(n-1)
The formula for a geometric sequence is: a(n)=a(1)*r^(n-1)

Now, when d is zero and r is one, a sequence is both geometric and arithmetic. This is because it becomes a(n)=a(1)1 =a(1).
It can easily observed that this makes the sequence a constant.

if a(1)=3 then for a geometric sequence a(n)=3+0(n-1)=3,3,3,3,3,3,3
and the geometric sequence a(n)=3r^0 = 3 also so the sequence is 3,3,3,3...
In fact, we could do this for any constant sequence such as 1,1,1,1,1,1,1... or e,e,e,e,e,e,e,e...