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 $S = \sum_{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

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) The given series is $S = \sum_{r=1}^n \log \left( \frac{a^r}{b^{r-1}} \right) = \log a + \log \left( \frac{a^2}{b} \right) + \log \left( \frac{a^3}{b^2} \right) + \dots + \log \left( \frac{a^n}{b^{n-1}} \right)$.This is an arithmetic progression $(A.P.)$ where the first term $a_1 = \log a$ and the $n$-th term $a_n = \log \left( \frac{a^n}{b^{n-1}} \right)$.The sum of $n$ terms of an $A.P.$ is given by $S_n = \frac{n}{2} (a_1 + a_n)$.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

Let $\alpha$ and $\beta$ be the roots of $x^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^{2}-6x+q=0$. If $\alpha, \beta, \gamma, \delta$ form a geometric progression,then the ratio $(2q+p):(2q-p)$ is:

For all positive integral values of $n$,the value of $3 \cdot 1 \cdot 2 + 3 \cdot 2 \cdot 3 + 3 \cdot 3 \cdot 4 + \dots + 3 \cdot n \cdot (n + 1)$ is

If $2^{10} + 2^{9} \cdot 3^{1} + 2^{8} \cdot 3^{2} + \ldots + 2^{0} \cdot 3^{10} = S - 2^{11}$,then $S$ is equal to

The first term of the $11^{th}$ group in the following sequence of groups $(1), (2, 3, 4), (5, 6, 7, 8, 9), \dots$ is:

If $a_1, a_2, \dots, a_n$ are in $A.P.$ with common difference $d$,then the sum of the following series is $\sin d (\csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \dots + \csc a_{n-1} \csc a_n)$

Vedclass Test Series

Exam Paper Generator

Online Exam Module