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Combined geometric sequences (Posted on 2024-11-02) Difficulty: 3 of 5
Let an, bn, and cn be geometric sequences with different common ratios and let an+bn+cn=dn for all integers n. If d1=1, d2=2, d3=3, d4=-7, d5=13 and d6=-16, find d7.

No Solution Yet Submitted by Danish Ahmed Khan    
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Possible answer Comment 1 of 1
Letting the terms of the geometric sequences be 
a, ar, ar^2 ...
b, bs, bs^2 ...
c, ct, ct^2 ...

Gives a system of 6 equations in 6 unknowns
a+b+c=1
ar+bs+ct=2 
...

Trying to solve this got unwieldy right away.  WolframAlpha balked at it, but was happier when I subbed 1-a-b in for c

solve ar+bs+(1-a-b)t=2 and ar^2+bs^2+(1-a-b)t^2=3 and ar^3+bs^3+(1-a-b)t^3=-7 and ar^4+bs^4+(1-a-b)t^4=13 and ar^5+bs^5+(1-a-b)t^5=-16

It gave 6 solutions (all permutations of a,b,c) one of them is

<img 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" alt="r≈-1.2328 - 0.7926 i and s≈0.46557 and t≈-1.2328 + 0.7926 i and b≈2.8694 and a≈-0.93472 - 1.03497 i">

These are rounded, so don't give the terms of d as 1,2,3,-7,13,-16 exactly but the real an imaginary parts were off by less than 10^-3.
So I'm pretty confident that when my calculator gave
ar^6+bs^6+cs^6 = 11.99822569 + 1.12116958x10^-4 i 
the actually answer is
d7 = 12.

  Posted by Jer on 2024-11-03 11:00:03
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