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Let the sum of the series a + a r 1 + a r 1 2 + . . . be r 1 , and the sum of the series a + a r 2 + a r 2 2 + . . . be r 2 . Both r 1 and r 2 are positive and r 1 = r 2 . What is the value of r 1 + r 2 ?
The answer is 1.000.
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1 solution
Editing the problem.
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I am a moderator and I have edited the problem. Because " a + a r + a r 2 + ⋯ to ∞ " could mean the last term tense to infinity. Formal math literature just ends the infinite series with ⋯ (see the examples in Wikipedia).
We are told that r 1 = 1 − r 1 a r 2 = 1 − r 2 a so that a = r 1 ( 1 − r 1 ) = r 2 ( 1 − r 2 ) Assuming that r 1 = r 2 , we deduce that r 1 , r 2 are the two roots of the quadratic equation X 2 − X + a = 0 , and hence r 1 + r 2 = 1 .
If we do not assume that r 1 = r 2 , we are a bit stuck, since we could have r 1 = r 2 where − 1 < r 1 < 1 and a = r 1 ( 1 − r 1 ) . Then r 1 + r 2 = 2 r 1 could take any value in ( − 2 , 2 ) .