(D) The $n^{th}$ term of a $G.P.$ is given by $l = a r^{n-1}$.Dividing both sides by $a$,we get $\frac{l}{a} = r^{n-1}$.Taking the logarithm on both s

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$1 + \frac{\log l - \log a}{\log r}$

(D) The $n^{th}$ term of a $G.P.$ is given by $l = a r^{n-1}$.Dividing both sides by $a$,we get $\frac{l}{a} = r^{n-1}$.Taking the logarithm on both sides,we have $\log(\frac{l}{a}) = \log(r^{n-1})$.Using the properties of logarithms,$\log l - \log a = (n-1) \log r$.Therefore,$n-1 = \frac{\log l - \log a}{\log r}$.Adding $1$ to both sides,we get $n = 1 + \frac{\log l - \log a}{\log r}$.

Class 11 Mathematics Sequences and Series Geometric progression

Easy

The first and last terms of a $G.P.$ are $a$ and $l$ respectively; $r$ being its common ratio; then the number of terms in this $G.P.$ is

A
$\frac{\log l - \log a}{\log r}$
B
$1 - \frac{\log l - \log a}{\log r}$
C
$\frac{\log a - \log l}{\log r}$
D
$1 + \frac{\log l - \log a}{\log r}$

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Sequences and Series

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Geometric progression

Let $C_0$ be a circle of radius $1$. For $n \geq 1$,let $C_n$ be a circle whose area equals the area of a square inscribed in $C_{n-1}$. Then,$\sum_{i=0}^{\infty} \text{Area}(C_i)$ equals

KVPY 2014

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For what values of $x$ are the numbers $\frac{2}{7}, x, \frac{7}{2}$ in $G.P.$?

Medium

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Which term of the $GP$ $2, 8, 32, \ldots$ is $131072$ (in $^{\text{th}}$)?

Easy

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Let $a_1, a_2, a_3, \ldots$ be a $GP$ of increasing positive numbers. If the product of the fourth and sixth terms is $9$ and the sum of the fifth and seventh terms is $24$,then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to $.........$.

DifficultJEE MAIN 2023

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If $\alpha, \beta, \gamma$ are the roots of the equation $3x^3-26x^2+52x-24=0$ such that $\alpha, \beta, \gamma$ are in geometric progression and $\alpha < \beta < \gamma$,then $3\alpha + 2\beta + \gamma =$

MediumTS EAMCET 2022

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The first and last terms of a $G.P.$ are $a$ and $l$ respectively; $r$ being its common ratio; then the number of terms in this $G.P.$ is

Similar Questions

Let $C_0$ be a circle of radius $1$. For $n \geq 1$,let $C_n$ be a circle whose area equals the area of a square inscribed in $C_{n-1}$. Then,$\sum_{i=0}^{\infty} \text{Area}(C_i)$ equals

For what values of $x$ are the numbers $\frac{2}{7}, x, \frac{7}{2}$ in $G.P.$?

Which term of the $GP$ $2, 8, 32, \ldots$ is $131072$ (in $^{\text{th}}$)?

Let $a_1, a_2, a_3, \ldots$ be a $GP$ of increasing positive numbers. If the product of the fourth and sixth terms is $9$ and the sum of the fifth and seventh terms is $24$,then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to $.........$.

If $\alpha, \beta, \gamma$ are the roots of the equation $3x^3-26x^2+52x-24=0$ such that $\alpha, \beta, \gamma$ are in geometric progression and $\alpha < \beta < \gamma$,then $3\alpha + 2\beta + \gamma =$

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