Find the sum: _ n=1 ^ 10 ( 12 )^ n-1 + ( 15 )^ n+1 . - Vidyadip

Question

Find the sum: $\sum\limits_{\text{n}=1}^{10}\bigg\{\Big(\frac12\Big)^{\text{n}-1}+\Big(\frac15\Big)^{\text{n}+1}\bigg\}.$

✓

Answer

$\sum\limits_{\text{n}=1}^{10}\bigg\{\Big(\frac12\Big)^{\text{n}-1}+\Big(\frac15\Big)^{\text{n}+1}\bigg\}$
$=\sum\limits_{\text{n}=1}^{10}\Big(\frac12\Big)^{\text{n}-1}+\sum\limits_{\text{n}=1}^{10}\Big(\frac15\Big)^{\text{n}+1}$
$=1+\frac12+\frac{1}{2^2}+\ \cdots\ +\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}+\ \cdots$
$=\frac{\Big(1-\frac{1}{2^{10}}\Big)}{1-\frac12}+\frac{\frac15\Big(1-\frac{1}{5^{10}}\Big)}{1-\frac15}$
$=\frac{2^{10}-1}{2^9}+\frac{5^{10}-1}{5^{11}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Geometric Progressions→Geometric Progressions — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow3}\big(\text{x}^2-9\big)\Big(\frac{1}{\text{x}+3}-\frac{1}{\text{x}-3}\Big)$

Find the equation of the straight lines passing through the following pair of points:
(a, b) and (a + b, a - b)

Find the equation of the line passing through the point of intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.

Solve the following linear inequations in R:
$\text{x}-2\leq\frac{5\text{x}+8}{3}$

Find the value of $n$ such that :
(i) ${ }^n P _5=42{ }^n P _3, n>4$
(ii) $\frac{{ }^n P _4}{{ }^{n-1} P _4}=\frac{5}{3}, n>4$

Find the sum of odd integers from 1 to 2001.

Find sin $\frac{x}{2},\cos \frac{x}{2}$ and $\tan \frac{x}{2}$ in the $\tan x = - \frac{4}{3}$, x in quadrant II.

Differentiate the following functions with respect to x:

$\frac{\sqrt{\text{a}}+\sqrt{\text{x}}}{\sqrt{\text{a}}-\sqrt{\text{x}}}$

Prove that $\frac{\tan\text{A}+\sec\text{A}-1}{\tan\text{A}-\sec\text{A}+1}=\frac{1+\sin\text{A}}{\cos\text{A}}$

Using binomial theorem write down the expansions of the following:

(1 - 3x)⁷