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

Algebra ->  Sequences-and-series -> 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       Log On


   

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) About Me  (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 a%5Bn%5D+=+12-2n+=+2%286-n%29 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 b%5Bn%5D+=+9%2A%282%2F3%29%5E%28n-1%29.
The gamma sequence is thus defined by the formula .
We have to find sum%2818%286-n%29%2A%282%2F3%29%5E%28n-1%29%2Cn+=+1%2C+infinity%29
= 18sum%28%286%2A%282%2F3%29%5E%28n-1%29-n%2A%282%2F3%29%5E%28n-1%29%29%2Cn+=+1%2C+infinity%29
=
=
The infinite geometric series 1%2Bx%2Bx%5E2%2Bx%5E3+... = 1%2F%281-x%29 as long as -1 < x < 1.
Staying within the domain of convergence and differentiating term-by-term, we get
1%2B2x%2B3x%5E2%2B4x%5E3+... = 1%2F%281-x%29%5E2

Now the first infinite sum is an infinite geometric series with sum 1%2F%281-2%2F3%29+=+1%2F%281%2F3%29+=+3.
The second infinite sum is derivative of the infinite geometric series and has sum 1%2F%281-2%2F3%29%5E2+=+1%2F%281%2F3%29%5E2+=+1%2F%281%2F9%29+=+9.
Therefore...

= 108*3 - 18*9 = highlight%28162%29.