SOLUTION: Alpha writes the infinite arithmetic sequence 10, 8, 6, 4, 2, 0 ... Beta writes the infinite geometric sequence 9, 6, 4, 8/3, 16/9 ... Gamma makes a sequence whose nth term is

Question 1036497: Alpha writes the infinite arithmetic sequence
10, 8, 6, 4, 2, 0 ...
Beta writes the infinite geometric sequence
9, 6, 4, 8/3, 16/9 ...
Gamma makes a sequence whose nth term is the product of the nth term of Alpha's sequence and the nth term of Beta's sequence:
10 * 9, 8 * 6, 6 * 4, 4 * 8/3, 2 * 16/9, ...
What is the sum of Gamma's entire sequence?
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
The alpha sequence is an arithmetic sequence with first term 10 and common difference -2, and is defined by the formula for n = 1,2,3,... .
The beta sequence is a geometric sequence with first term 9 and common ratio 2/3, and is defined by the formula .
The gamma sequence is thus defined by the formula .
We have to find
=
=
=
The infinite geometric series +... = as long as -1 < x < 1.
Staying within the domain of convergence and differentiating term-by-term, we get
+... =

Now the first infinite sum is an infinite geometric series with sum .
The second infinite sum is derivative of the infinite geometric series and has sum .
Therefore...

= 108*3 - 18*9 = .

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