Friday, 21 June 2013

sequences and series - Geometric progression — Sum of terms, sum of terms' cubes, sum of terms' squares



Consider the infinite geometric progression a1,a2,a3,. Given the sum of its terms, as well as the sum of the terms' cubes



a1+a2+a3+=3a31+a32+a33+=10813



find the sum of the terms' squares




a21+a22+a23+






This is what I have so far:



a1+a2+a3+=31+q+q2+=3a1



a31+a32+a33+=108131+q3+q6+=10813a1



a21+a22+a23+=a31a1+a32a2+a33a3+=a31a1(1+q3q+q6q2+)



Any help will be much appreciated.


Answer



HINT:




Let the first term be a and the common ratio be r



For convergence we need |r|<1



So, the sum of the first series 0n<arn=a1r=3    (1)



So, the sum of the second series 0n<a3(r3)n=a31r3=10813    (2)



Cube the first & divide by (2) to get r and then get a from (1)




We need 0n<a2(r2)n=a21r2


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