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Standard XII

Mathematics

Infinite GP

Let a + ar 1+...

Question

Let $a + a r_{1} + a {r_{1}}^{2} + . . .$ and $a + a r_{2} + a {r_{2}}^{2} + . . .$ be two infinite series of positive numbers with the same first term, where $r_{1}, r_{2} < 1$ . The sum of the first series is $r_{1}$ and the sum of the second series is $r_{2}$ then the value of $r_{1} + r_{2}$ is

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Solution

$a + a r_{1} + a {r_{1}}^{2} + . . . \infty = r_{1} a > 0$
$a + a r_{2} + a {r_{2}}^{2} + . . . \infty = r_{2} 1 > r_{1}, r_{2} > 0$
Let
$S (r) = a + a r + a r^{2} + . . . \infty = r$
$\Rightarrow \frac{a}{1 - r} = r$
$\Rightarrow r^{2} - r + a = 0$
So the above equation roots will be $r_{1}, r_{2}$
Sum of the roots
$\Rightarrow r_{1} + r_{2} = 1$

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Let a+ar1+ar12+... and a+ar2+ar22+... be two infinite series of positive numbers with the same first term, where r1,r2<1. The sum of the first series is r1 and the sum of the second series is r2 then the value of r1+r2 is

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Let $a + a r_{1} + a {r_{1}}^{2} + . . .$ and $a + a r_{2} + a {r_{2}}^{2} + . . .$ be two infinite series of positive numbers with the same first term, where $r_{1}, r_{2} < 1$ . The sum of the first series is $r_{1}$ and the sum of the second series is $r_{2}$ then the value of $r_{1} + r_{2}$ is