Download App

Sequences and Series — Applied Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceApplied MathsSequences and Series5 Marks
Question
If $x=a+\frac{a}{r}+\frac{a}{r^2}+\ldots+\infty, y=b-\frac{b}{r}+\frac{b}{r^2}-\ldots+\infty$ and $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\ldots+\infty$ Prove that $\frac{x . y}{z}=\frac{a b}{c}$
✓

Answer

Given series $x=a+\frac{a}{r}+\frac{a}{r^2}+\ldots+\infty$ is in G.P. with common ratio $\frac{1}{r}$.
and series $y=b-\frac{b}{r}+\frac{b}{r^2}-\ldots+\infty$ is in G.P. with common ratio $-\frac{1}{r}$.
and series $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\ldots+\infty$ is in G.P. with common ratio $\frac{1}{r^2}$
Sum of infinite terms of series$
x=a+\frac{a}{r}+\frac{a}{r^2}+\ldots+\infty \text { is } x=\frac{a}{1-\frac{1}{r}}
$
Sum of infinite term of series $y=b-\frac{b}{r}+\frac{b}{r^2}-\ldots+\infty$ is
$
y=\frac{b}{1-\left(-\frac{1}{r}\right)}=\frac{b}{1+\frac{1}{r}}
$
Sum of infinite terms of series $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\ldots+\infty$ is
$
z=\frac{c}{1-\frac{1}{r^2}}
$
Now, LHS $=\frac{x y}{z}$
$=\frac{\left\{\frac{a}{\left(1-\frac{1}{r}\right)}\right\}\left\{\frac{b}{\left(\left(1+\frac{1}{r}\right)\right.}\right\}}{\frac{c}{1-\frac{1}{r^2}}}$
$\begin{array}{l}=\frac{\frac{a b}{\left(1-\frac{1}{r^2}\right)}}{\frac{c}{\left(1-\frac{1}{r^2}\right)}} \\ =\frac{a b}{c}=\text { R.H.S }\end{array}$
Hence Proved

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 "5 Marks Questions" from Sequences and Series→Sequences and Series — all question types→Applied Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Coefficient of correlation between $x$ and $y$ for 20 items is 0.4. The AM's and SD's of x and y are known to be 12 and 15 and 3 and 4, re Later on, it was found that the pair (2 wrongly taken as (15, 20). Find the correct value of the correlation coefficient.
→
Find the Spearman's rank correlation between marks in Mathematics and Statistics obtained by 10 students:
Maths in Mathematics80389530748491606640
Marks in Statistics85509258706588565246
→
How many 3-digit numbers are there which have exactly one of their digits as 6 ?
→
Find the equations of the lines through the point $(3,2)$ which make an angle of $45^{\circ}$ with the line $\mathrm{x}-2 \mathrm{y}=3$.
→
Evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$
→
A person opened an account on April, 2019 with a deposit of 800. The account paid 6% interest compounded quarterly. On October 1, 2019 he closed the account and added enough additional money to invest in a 6 month time-deposit for ₹ 1000, earning 6% compounded monthly.
(a) How much addition amount did the person invest on October 1?
(b) What was the maturity value of his time deposit on April 1 2002 ?
(c) How much total interest was earned ?
Given that $(1+i)^n$ is 1.03022500 for $i=1 \frac{1}{2}, n=2$
and $(1+i)^n$ is 1.03037751 for $i=\frac{1}{2} \%, n=6$.
→
In how many ways can 9 examination papers be arranged so that the best and the worst papers are never together?
→
If $f(x)=\left\{\begin{array}{lc}m x^2+n, & x<0 \\ n x+m, & 0 \leq x \leq 1 . \\ n x^3+m, & x>1\end{array}\right.$For what integers $m$ and $n$ does both $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow 1} f(x)$ exist ?
→
Divide 32 into four parts which are in AP such that the product of extremes is to the product of means as $7: 15$.
→
Find the equation of a straight line which passes through the point $(a, 0)$ and whose perpendicular distance from $(2 a, 2 a)$ is $a$.
→