What is this geometric sequence? Sum of first few terms GP is 11 , the sum of squares is 341 and sum of cubes is 3641.

Question

What is this geometric sequence?
The sum of first few terms geometric sequence is 11, the sum of squares is 341 and sum of cubes is 3641.

Answer 1

1,-2,4,-8,16 and 16,-8,4,-2,1

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a=first term, q=common ratio, n=number of terms.
Q=qn.
q=1 is impossible, as in that case an=11, a2n=341, and hence a=31, n=11/31, which contradicts to the condition a3n=3641
Therefore, the following relationships are true:
(1)-> a{Q-1 / q-1}=11
(2)-> a^2{Q^2-1 / q^2-1}=341
(3)-> a^3{Q^3-1 /r q^3-1}=3641
Divide (2) and (3) by (1).
Then a{Q+1 / q+1}=31 a^2{Q^2+Q+1 / q^2+q+-1}=331.
Hence a(Q-1)=11(q-1) (Q+1)=31(q+1) thus, a=10q+21 and Q={21q+10 / 10q+21}
Substitute the result into the equality a2(Q2+Q+1)=331(q2+q+1).
possible values of q : the roots of the equation 2q2+5q+2=0: q=-2 and q=-1/2.
Hence either a=1, Q=-32, n=5, or a=16, Q=-1/32, n=5.
Thus the solutions of the problem are the following two sequences: 1,-2,4,-8,16 and 16,-8,4,-2,1