(A) The given series is an infinite geometric progression $(G.P.)$ with the first term $a = x$ and common ratio $r = x$.For an infinite $G.P.$ to conv

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(A) The given series is an infinite geometric progression $(G.P.)$ with the first term $a = x$ and common ratio $r = x$.For an infinite $G.P.$ to converge,$|r| The sum of an infinite $G.P.$ is given by the formula $S = \frac{a}{1 - r}$.Substituting the values,we get $y = \frac{x}{1 - x}$.Now,solve for $x$:$y(1 - x) = x$$y - yx = x$$y = x + yx$$y = x(1 + y)$$x = \frac{y}{1 + y}$.

Competitive Quantitative Aptitude Progression and Sequence Progression and Sequence

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If $y = x + x^2 + x^3 + \dots \infty$,then $x = $

A
$\frac{y}{1 + y}$
B
$\frac{1 - y}{y}$
C
$\frac{y}{1 - y}$
D
None of these

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Progression and Sequence

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Progression and Sequence

The sum of the series $1 \cdot 3^2 + 2 \cdot 5^2 + 3 \cdot 7^2 + \dots$ up to $20$ terms is

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Let ${S_n}$ denote the sum of $n$ terms of an $A.P.$ If ${S_{2n}} = 3{S_n}$,then the ratio $\frac{{{S_{3n}}}}{{{S_n}}}$ is equal to:

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The first term of an $A.P.$ of consecutive integers is $p^2 + 1$. The sum of $(2p + 1)$ terms of this series can be expressed as:

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If $1^2 + 2^2 + 3^2 + \dots + 2009^2 = (2009)(335)(4019)$ and $(1)(2009) + 2(2008) + 3(2007) + \dots + 2009(1) = (2009)(335)(x)$,then $x$ is equal to:

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The terms of a $G.P.$ are positive. If each term is equal to the sum of two terms that follow it,then the common ratio is

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If $y = x + x^2 + x^3 + \dots \infty$,then $x = $

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The sum of the series $1 \cdot 3^2 + 2 \cdot 5^2 + 3 \cdot 7^2 + \dots$ up to $20$ terms is

Let ${S_n}$ denote the sum of $n$ terms of an $A.P.$ If ${S_{2n}} = 3{S_n}$,then the ratio $\frac{{{S_{3n}}}}{{{S_n}}}$ is equal to:

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If $1^2 + 2^2 + 3^2 + \dots + 2009^2 = (2009)(335)(4019)$ and $(1)(2009) + 2(2008) + 3(2007) + \dots + 2009(1) = (2009)(335)(x)$,then $x$ is equal to:

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