If for x, y in mathbb{R}, x 0, y = log_{10} | vedclass

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

Question

(D) Given the expression for $y$:$y = (\log_{10} x) (1 + \frac{1}{3} + \frac{1}{9} + \dots \infty)$Using the sum of an infinite geometric series $S =

Vedclass · Accepted Answer

$(10^6, 9)$

Vedclass · Answer

(D) Given the expression for $y$:$y = (\log_{10} x) (1 + \frac{1}{3} + \frac{1}{9} + \dots \infty)$Using the sum of an infinite geometric series $S = \frac{a}{1-r}$ where $a=1$ and $r=\frac{1}{3}$:$y = (\log_{10} x) \left( \frac{1}{1 - 1/3} ight) = (\log_{10} x) \left( \frac{3}{2} ight) = \frac{3}{2} \log_{10} x$Now,consider the given equation:$\frac{2(1+2+3+\dots+y)}{3(1+2+3+\dots+y)} = \frac{4}{\log_{10} x}$$\frac{2}{3} = \frac{4}{\log_{10} x}$$\log_{10} x = \frac{4 	imes 3}{2} = 6$$x = 10^6$Substitute $\log_{10} x = 6$ into the expression for $y$:$y = \frac{3}{2} (6) = 9$Thus,the ordered pair $(x, y)$ is $(10^6, 9)$.

If for $x, y \in \mathbb{R}, x > 0,$ $y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \dots$ up to $\infty$ terms and $\frac{2+4+6+\dots+2y}{3+6+9+\dots+3y} = \frac{4}{\log_{10} x}$,then the ordered pair $(x, y)$ is equal to:

Similar Questions

Let $3, 6, 9, 12, \ldots$ up to $78$ terms and $5, 9, 13, 17, \ldots$ up to $59$ terms be two series. Then,the sum of the terms common to both the series is equal to

If the value of $\left(1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots \text{ to } \infty\right)^{\log_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \text{ to } \infty\right)}$ is $l$,then $l^{2}$ is equal to $......$

Let $a_1=b_1=1$ and $a_n=a_{n-1}+(n-1)$,$b_n=b_{n-1}+a_{n-1}$,$\forall n \geq 2$. If $S =\sum \limits_{n=1}^{10} \frac{b_n}{2^n}$ and $T =\sum \limits_{n=1}^8 \frac{n}{2^{n-1}}$,then $2^7(2S - T)$ is equal to $........$.

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = (-1)^{n-1} 5^{n+1}$.

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. The greatest value of $n$ for which $a_n$ is the greatest is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module