Value of Σlimits_{r = 1}^n {log left( {frac{{{a^r}}}{{{b^{r

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8. Sequences and Series

medium

The value of $\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} $ is

$\frac{n}{2}\log \left( {\frac{{{a^n}}}{{{b^n}}}} \right)$

$\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^n}}}} \right)$

$\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n - 1}}}}} \right)$

$\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n + 1}}}}} \right)$

Solution

$\log a + \log \left( {\frac{{{a^2}}}{b}} \right) + \log \left( {\frac{{{a^3}}}{{{b^2}}}} \right) + \log \left( {\frac{{{a^4}}}{{{b^3}}}} \right) + …… + \log \left( {\frac{{{a^n}}}{{{b^{n – 1}}}}} \right)$

This is an $A.P.$ with first term $\log a$

and the common difference $\log \left( {\frac{{{a^2}}}{b}} \right) – \log a = \log \left( {\frac{a}{b}} \right)$

Therefore the sum of $n$ terms is

$\frac{n}{2}\left[ {\log a + \log \left( {\frac{{{a^n}}}{{{b^{n – 1}}}}} \right)} \right] = \frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n – 1}}}}} \right)$.

Trick : Check for $n = 1,\;2$.

Standard 11

Mathematics

Let $AP ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $d >0$. If $\operatorname{AP}(1 ; 3) \cap \operatorname{AP}(2 ; 5) \cap \operatorname{AP}(3 ; 7)=\operatorname{AP}( a ; d )$ then $a + d$ equals. . . . .

normal

(IIT-2019)

If the first term of an $A.P.$ is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first $20$ terms is equal to

hard

(JEE MAIN-2025)

The arithmetic mean of first $n$ natural number

easy

Let $s _1, s _2, s _3, \ldots \ldots, s _{10}$ respectively be the sum to 12 terms of 10 A.P.s whose first terms are $1,2,3, \ldots, 10$ and the common differences are $1,3,5, \ldots, 19$ respectively. Then $\sum \limits_{i=1}^{10} s _{ i }$ is equal to

hard

(JEE MAIN-2023)

The sixth term of an $A.P.$ is equal to $2$, the value of the common difference of the $A.P.$ which makes the product ${a_1}{a_4}{a_5}$ least is given by

hard

The value of $\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} $ is

Solution

Similar Questions

Let $AP ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $d >0$. If $\operatorname{AP}(1 ; 3) \cap \operatorname{AP}(2 ; 5) \cap \operatorname{AP}(3 ; 7)=\operatorname{AP}( a ; d )$ then $a + d$ equals. . . . .

If the first term of an $A.P.$ is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first $20$ terms is equal to

The arithmetic mean of first $n$ natural number

Let $s _1, s _2, s _3, \ldots \ldots, s _{10}$ respectively be the sum to 12 terms of 10 A.P.s whose first terms are $1,2,3, \ldots, 10$ and the common differences are $1,3,5, \ldots, 19$ respectively. Then $\sum \limits_{i=1}^{10} s _{ i }$ is equal to

The sixth term of an $A.P.$ is equal to $2$, the value of the common difference of the $A.P.$ which makes the product ${a_1}{a_4}{a_5}$ least is given by

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