The value of sumlimits_{r = 1}^n {log left( { | vedclass

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

Question

(C) The given series is $\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} = \log a + \log \left( {\frac{{{a^2}}}{b}} \right

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) The given series is $\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} = \log a + \log \left( {\frac{{{a^2}}}{b}} \right) + \log \left( {\frac{{{a^3}}}{{{b^2}}}} \right) + \dots + \log \left( {\frac{{{a^n}}}{{{b^{n - 1}}}}} \right)$.This is an arithmetic progression $(A.P.)$ where the first term $A = \log a$ and the last term $L = \log \left( {\frac{{{a^n}}}{{{b^{n - 1}}}}} \right)$.The sum of $n$ terms of an $A.P.$ is given by $S_n = \frac{n}{2}(A + L)$.Substituting the values,we get $S_n = \frac{n}{2} \left[ \log a + \log \left( {\frac{{{a^n}}}{{{b^{n - 1}}}}} \right) \right]$.Using the property $\log x + \log y = \log(xy)$,we have $S_n = \frac{n}{2} \log \left( a \cdot \frac{{{a^n}}}{{{b^{n - 1}}}} \right) = \frac{n}{2} \log \left( {\frac{{{a^{n + 1}}}}{{{b^{n - 1}}}}} \right)$.

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

Similar Questions

If the arithmetic mean between the $p^{th}$ term and $q^{th}$ term of an arithmetic progression is equal to the arithmetic mean between its $r^{th}$ and $s^{th}$ term,then $p + q = ......$

If $a_1, a_2, \dots, a_n$ are positive real numbers such that their product is a constant $c$,then what is the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$?

If ${a_1}, {a_2}, {a_3}, \dots, {a_{24}}$ are in arithmetic progression and ${a_1} + {a_5} + {a_{10}} + {a_{15}} + {a_{20}} + {a_{24}} = 225$,then ${a_1} + {a_2} + {a_3} + \dots + {a_{23}} + {a_{24}} = $

Let $a_1, a_2, a_3, \ldots, a_n$ be positive real numbers. Then the minimum value of $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\ldots+\frac{a_n}{a_1}$ is

For $x \geq 0$,the least value of $K$,for which $4^{1+x}+4^{1-x}$,$\frac{K}{2}$,and $16^{x}+16^{-x}$ are three consecutive terms of an $A.P.$ is equal to :

Vedclass Test Series

Exam Paper Generator

Online Exam Module