Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

If $open vertical bar a close vertical bar less than 1$ and $open vertical bar b close vertical bar less than 1$ , then the sum of the series $1 plus open parentheses 1 plus a close parentheses b plus open parentheses 1 plus a plus a to the power of 2 end exponent close parentheses b to the power of 2 end exponent plus open parentheses 1 plus a plus a to the power of 2 end exponent plus a to the power of 3 end exponent close parentheses b to the power of 3 end exponent plus horizontal ellipsis$ is

$\frac{1}{(1 - a) (1 - b)}$
$\frac{1}{(1 - a) (1 - a b)}$
$\frac{1}{(1 - b) (1 - a b)}$
$\frac{1}{(1 - a) (1 - b) (1 - a b)}$

The correct answer is: $fraction numerator 1 over denominator open parentheses 1 minus b close parentheses left parenthesis 1 minus a b right parenthesis end fraction$

We have,
$1 plus open parentheses 1 plus a close parentheses b plus open parentheses 1 plus a plus a to the power of 2 end exponent close parentheses b to the power of 2 end exponent plus open parentheses 1 plus a plus a to the power of 2 end exponent plus a to the power of 3 end exponent close parentheses b to the power of 3 end exponent plus horizontal ellipsis infinity blank$
$equals not stretchy sum from n equals 1 to infinity of left parenthesis 1 plus a plus a to the power of 2 end exponent plus horizontal ellipsis plus a to the power of n minus 1 end exponent right parenthesis b to the power of n minus 1 end exponent$
$equals not stretchy sum from n equals 1 to infinity of open parentheses fraction numerator 1 minus a to the power of n end exponent over denominator 1 minus a end fraction close parentheses b to the power of n minus 1 end exponent$
$equals not stretchy sum from n equals 1 to infinity of fraction numerator b to the power of n minus 1 end exponent over denominator 1 minus a end fraction minus not stretchy sum from n equals 1 to infinity of fraction numerator a to the power of n end exponent b to the power of n minus 1 end exponent over denominator 1 minus a end fraction$
$equals fraction numerator 1 over denominator 1 minus a end fraction not stretchy sum from n equals 1 to infinity of b to the power of n minus 1 end exponent minus fraction numerator a over denominator 1 minus a end fraction not stretchy sum from n equals 1 to infinity of open parentheses a b close parentheses to the power of n minus 1 end exponent$
$equals fraction numerator 1 over denominator 1 minus a end fraction open square brackets 1 plus b plus b to the power of 2 end exponent plus horizontal ellipsis infinity close square brackets minus fraction numerator a over denominator 1 minus a end fraction left square bracket 1 plus a b plus open parentheses a b close parentheses to the power of 2 end exponent plus horizontal ellipsis infinity right square bracket$
$equals fraction numerator 1 over denominator 1 minus a end fraction cross times fraction numerator 1 over denominator 1 minus b end fraction minus fraction numerator a over denominator open parentheses 1 minus a close parentheses left parenthesis 1 minus a b right parenthesis end fraction$
$equals fraction numerator 1 over denominator open parentheses 1 minus a b close parentheses left parenthesis 1 minus b right parenthesis end fraction$

Related Questions to study

The value of $not stretchy sum from r equals 0 to n of open parentheses a plus r plus a r close parentheses open parentheses negative a close parentheses to the power of r end exponent$ is equal to

The value of $not stretchy sum from r equals 0 to n of open parentheses a plus r plus a r close parentheses open parentheses negative a close parentheses to the power of r end exponent$ is equal to

The sum of series $1 plus fraction numerator 4 over denominator 5 end fraction plus fraction numerator 7 over denominator 5 to the power of 2 end exponent end fraction plus fraction numerator 10 over denominator 5 to the power of 3 end exponent end fraction plus horizontal ellipsis infinity$ is

The sum of series $1 plus fraction numerator 4 over denominator 5 end fraction plus fraction numerator 7 over denominator 5 to the power of 2 end exponent end fraction plus fraction numerator 10 over denominator 5 to the power of 3 end exponent end fraction plus horizontal ellipsis infinity$ is

If $t subscript n end subscript equals fraction numerator 1 over denominator 4 end fraction open parentheses n plus 2 close parentheses left parenthesis n plus 3 right parenthesis$ for $n equals 1 comma blank 2 comma blank 3 comma blank horizontal ellipsis comma blank$ then $fraction numerator 1 over denominator t subscript 1 end subscript end fraction plus fraction numerator 1 over denominator t subscript 2 end subscript end fraction plus fraction numerator 1 over denominator t subscript 3 end subscript end fraction plus horizontal ellipsis plus fraction numerator 1 over denominator t subscript 2003 end subscript end fraction equals$

If $t subscript n end subscript equals fraction numerator 1 over denominator 4 end fraction open parentheses n plus 2 close parentheses left parenthesis n plus 3 right parenthesis$ for $n equals 1 comma blank 2 comma blank 3 comma blank horizontal ellipsis comma blank$ then $fraction numerator 1 over denominator t subscript 1 end subscript end fraction plus fraction numerator 1 over denominator t subscript 2 end subscript end fraction plus fraction numerator 1 over denominator t subscript 3 end subscript end fraction plus horizontal ellipsis plus fraction numerator 1 over denominator t subscript 2003 end subscript end fraction equals$

parallel

The sum to 50 terms of the series $1 plus 2 open parentheses 1 plus fraction numerator 1 over denominator 50 end fraction close parentheses plus 3 open parentheses 1 plus fraction numerator 1 over denominator 50 end fraction close parentheses to the power of 2 end exponent plus horizontal ellipsis$ is given by

The sum to 50 terms of the series $1 plus 2 open parentheses 1 plus fraction numerator 1 over denominator 50 end fraction close parentheses plus 3 open parentheses 1 plus fraction numerator 1 over denominator 50 end fraction close parentheses to the power of 2 end exponent plus horizontal ellipsis$ is given by

The line $x plus y equals 1$ meets $x$ -axis at $A$ and $y$ -axis at $B comma blank P$ is the mid-point of $A B$ ;
$P subscript 1 end subscript$ is the foot of the perpendicular from $P$ to $O A$ :
$M subscript 1 end subscript$ is that of $P subscript 1 end subscript$ from $O P$ ; $P subscript 2 end subscript$ is that of $M subscript 1 end subscript$ from $O A$ ; $M subscript 2 end subscript$ is that of $P subscript 2 end subscript$ from $O P$ ; $P subscript 3 end subscript$ is that of $M subscript 2 end subscript$ from $O A$ ; and so on.
If $P subscript n end subscript$ denotes the $n to the power of t h end exponent$ foot of the perpendicular on $O A$ : then $O P subscript n end subscript$ is

The line $x plus y equals 1$ meets $x$ -axis at $A$ and $y$ -axis at $B comma blank P$ is the mid-point of $A B$ ;
$P subscript 1 end subscript$ is the foot of the perpendicular from $P$ to $O A$ :
$M subscript 1 end subscript$ is that of $P subscript 1 end subscript$ from $O P$ ; $P subscript 2 end subscript$ is that of $M subscript 1 end subscript$ from $O A$ ; $M subscript 2 end subscript$ is that of $P subscript 2 end subscript$ from $O P$ ; $P subscript 3 end subscript$ is that of $M subscript 2 end subscript$ from $O A$ ; and so on.
If $P subscript n end subscript$ denotes the $n to the power of t h end exponent$ foot of the perpendicular on $O A$ : then $O P subscript n end subscript$ is
Maths-General

Let $a element of left parenthesis 0 comma blank 1 right square bracket$ satisfies the equation $a to the power of 2008 end exponent minus 2 a plus 1 equals 0$ and $S equals 1 plus a plus a to the power of 2 end exponent plus... plus a to the power of 2007 end exponent$ . Sum of all possible value(s) of $S$ , is

Let $a element of left parenthesis 0 comma blank 1 right square bracket$ satisfies the equation $a to the power of 2008 end exponent minus 2 a plus 1 equals 0$ and $S equals 1 plus a plus a to the power of 2 end exponent plus... plus a to the power of 2007 end exponent$ . Sum of all possible value(s) of $S$ , is

parallel

The maximum sum of the series $20 plus 19 fraction numerator 1 over denominator 3 end fraction plus 18 fraction numerator 2 over denominator 3 end fraction plus horizontal ellipsis$ is

The maximum sum of the series $20 plus 19 fraction numerator 1 over denominator 3 end fraction plus 18 fraction numerator 2 over denominator 3 end fraction plus horizontal ellipsis$ is

The sum of an infinite G.P. is 57 and the sum of their cubes is 9747, then common ratio of the G.P. is

The sum of an infinite G.P. is 57 and the sum of their cubes is 9747, then common ratio of the G.P. is

If $a comma blank b comma$ and $c$ are in A.P. and $b minus a comma blank c minus b$ and $a$ are in G.P., then $a colon b colon c$ is

If $a comma blank b comma$ and $c$ are in A.P. and $b minus a comma blank c minus b$ and $a$ are in G.P., then $a colon b colon c$ is

parallel

The coefficient of $x to the power of 49 end exponent$ in the product $open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses midline horizontal ellipsis left parenthesis x minus 99 right parenthesis$ is

The coefficient of $x to the power of 49 end exponent$ in the product $open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses midline horizontal ellipsis left parenthesis x minus 99 right parenthesis$ is

If $t subscript n end subscript$ denotes the $n to the power of t h end exponent$ term of the series $2 plus 3 plus 6 plus 11 plus 18 plus horizontal ellipsis$ then $t subscript 50 end subscript$ is

If $t subscript n end subscript$ denotes the $n to the power of t h end exponent$ term of the series $2 plus 3 plus 6 plus 11 plus 18 plus horizontal ellipsis$ then $t subscript 50 end subscript$ is

An infinite GP has first term $x$ and sum 5, then

An infinite GP has first term $x$ and sum 5, then

parallel

If $a comma blank b$ and $c$ are in A.P. then $a to the power of 3 end exponent plus c to the power of 3 end exponent minus 8 b to the power of 3 end exponent$ is equal to

If $a comma blank b$ and $c$ are in A.P. then $a to the power of 3 end exponent plus c to the power of 3 end exponent minus 8 b to the power of 3 end exponent$ is equal to

$A B C$ is a right-angled triangle in which $angle B equals 90 degree$ and $B C equals a$ . If $n$ points $L subscript 1 end subscript comma L subscript 2 end subscript comma horizontal ellipsis comma L subscript n end subscript$ on $A B$ is divided in $n plus 1$ equals parts and $L subscript 1 end subscript M subscript 1 end subscript comma L subscript 2 end subscript M subscript 2 end subscript comma blank horizontal ellipsis comma blank L subscript n end subscript M subscript n end subscript$ are line segments parallel to $B C$ and $M subscript 1 end subscript comma M subscript 2 end subscript comma horizontal ellipsis comma M subscript n end subscript$ are on $A C$ , then the sum of the lengths of $L subscript 1 end subscript M subscript 1 end subscript comma blank L subscript 2 end subscript M subscript 2 end subscript comma horizontal ellipsis comma L subscript n end subscript M subscript n end subscript$ is

$A B C$ is a right-angled triangle in which $angle B equals 90 degree$ and $B C equals a$ . If $n$ points $L subscript 1 end subscript comma L subscript 2 end subscript comma horizontal ellipsis comma L subscript n end subscript$ on $A B$ is divided in $n plus 1$ equals parts and $L subscript 1 end subscript M subscript 1 end subscript comma L subscript 2 end subscript M subscript 2 end subscript comma blank horizontal ellipsis comma blank L subscript n end subscript M subscript n end subscript$ are line segments parallel to $B C$ and $M subscript 1 end subscript comma M subscript 2 end subscript comma horizontal ellipsis comma M subscript n end subscript$ are on $A C$ , then the sum of the lengths of $L subscript 1 end subscript M subscript 1 end subscript comma blank L subscript 2 end subscript M subscript 2 end subscript comma horizontal ellipsis comma L subscript n end subscript M subscript n end subscript$ is

Consider an infinite geometric series with first term $a$ and common ratio $r$ . If its sum is 4 and the second term is 3/4, then

Consider an infinite geometric series with first term $a$ and common ratio $r$ . If its sum is 4 and the second term is 3/4, then

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.