If x = sum_{n=0}^{infty} a^{n},y = sum_{n=0}^ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $x = \sum_{n=0}^{\infty} a^{n} = \frac{1}{1-a}$,$y = \frac{1}{1-b}$,and $z = \frac{1}{1-c}$.From these,we get $a = 1 - \frac{1}{x}$,$b = 1 -

Vedclass · Accepted Answer

$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in $A.P.$

Vedclass · Answer

(B) Given $x = \sum_{n=0}^{\infty} a^{n} = \frac{1}{1-a}$,$y = \frac{1}{1-b}$,and $z = \frac{1}{1-c}$.From these,we get $a = 1 - \frac{1}{x}$,$b = 1 - \frac{1}{y}$,and $c = 1 - \frac{1}{z}$.Since $a, b, c$ are in $A.P.$,we have $2b = a + c$.Substituting the expressions in terms of $x, y, z$:$2(1 - \frac{1}{y}) = (1 - \frac{1}{x}) + (1 - \frac{1}{z})$$2 - \frac{2}{y} = 2 - (\frac{1}{x} + \frac{1}{z})$$\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$.This condition implies that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in $A.P.$

If $x = \sum_{n=0}^{\infty} a^{n}$,$y = \sum_{n=0}^{\infty} b^{n}$,$z = \sum_{n=0}^{\infty} c^{n}$,where $a, b, c$ are in $A.P.$ and $|a| < 1, |b| < 1, |c| < 1$,$abc \neq 0$,then:

Similar Questions

If $a, b, c$ are three unequal numbers such that $a, b, c$ are in $A.P.$ and $b - a, c - b, a$ are in $G.P.$,then $a : b : c$ is

The number of terms common between the two series $2 + 5 + 8 + \dots$ up to $50$ terms and the series $3 + 5 + 7 + 9 + \dots$ up to $60$ terms is:

If the arithmetic mean of two numbers $a$ and $b$ is twice their geometric mean,then $a : b = \dots$

Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{a}, \frac{1}{b}, 6$ be in $A.P.,$ where $a, b > 0$. Then $72(a + b)$ is equal to ...... .

Given an $A.P.$ and a $G.P.$ with positive terms, where the first and second terms of both progressions are equal. If $a_n$ and $b_n$ are the $n^{\text{th}}$ terms of the $A.P.$ and $G.P.$ respectively, then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module