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+⋯=3⟹1+q+q2+⋯=3a1
a31+a32+a33+⋯=10813⟹1+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 ∑0≤n<∞a⋅rn=a1−r=3 (1)
So, the sum of the second series ∑0≤n<∞a3⋅(r3)n=a31−r3=10813 (2)
Cube the first & divide by (2) to get r and then get a from (1)
We need ∑0≤n<∞a2⋅(r2)n=a21−r2
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