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Please help: the sum of the first two terms of a G.P is x.The sum of the last two is 1.
if there are n term in all, calculate the common ratio
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Let the GP be a, ar, ar^2, . . . , ar^(n-1).
The first term is "a", the common ratio is "r"; the number of terms is "n".
We are given
(a) the value of n; (the number of terms);
(b) a + ar = x; (the sum of the 1st and the 2nd terms, x)
(c) ar^(n-2) + ar^(n-1) = 1. (the sum of the two last terms, 1).
From (a), we have
a*(1+r) = x. (1)
From (b), we have
= 1. (2)
Now divide equation (2) by equation (1). You will get, after canceling common factors
= .
It implies
r = = = . ANSWER
Any of the equivalent forms in the line above is the answer.
So, the value of the common ratio "r" is by the input data x and n.
Solved.
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As a conclusion
The problem is posed correctly. It has (it admits) a unique solution.
Tutor @math_tutor2020 treated the problem incorrectly. He missed part of input data.
Therefore, his solution is incorrect.
His diagnosis
"there are infinitely many possible solutions here.
There isn't enough info to answer this current question."
is incorrect.
The problem is posed correctly, and I gave a complete solution/analysis.