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

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

1/3
2/3
1/6
None of these

The correct answer is: 2/3

Let $a$ be the first term and $r$ the common ratio of the G.P. Then, the sum is given by
$fraction numerator a over denominator 1 minus r end fraction equals 57$ (1)
Sum of the cubes is 9747. Hence,
$a to the power of 3 end exponent plus a to the power of 3 end exponent r to the power of 3 end exponent plus a to the power of 3 end exponent r to the power of 6 end exponent plus horizontal ellipsis equals 9747$
$rightwards double arrow fraction numerator a to the power of 3 end exponent over denominator 1 minus r to the power of 3 end exponent end fraction equals 9747$ (2)
Dividing the cube of (1) by (2), we get
$fraction numerator a to the power of 3 end exponent over denominator open parentheses 1 minus r close parentheses to the power of 3 end exponent end fraction fraction numerator left parenthesis 1 minus r to the power of 3 end exponent right parenthesis over denominator a to the power of 3 end exponent end fraction equals fraction numerator open parentheses 57 close parentheses to the power of 3 end exponent over denominator 9747 end fraction$
$rightwards double arrow fraction numerator 1 minus r to the power of 3 end exponent over denominator open parentheses 1 minus r close parentheses to the power of 3 end exponent end fraction equals 19$
$rightwards double arrow fraction numerator 1 plus r plus r to the power of 2 end exponent over denominator open parentheses 1 minus r close parentheses to the power of 2 end exponent end fraction equals 19$
$rightwards double arrow 18 r to the power of 2 end exponent minus 39 r plus 18 equals 0$
$rightwards double arrow open parentheses 3 r minus 2 close parentheses open parentheses 6 r minus 9 close parentheses equals 0$
$rightwards double arrow r equals 2 divided by 3$ or $r equals 3 divided by 2$
$equals 2 divided by 3$ [ $because blank r not equal to 3 divided by 2$ , because $0 less than open vertical bar r close vertical bar less than 1$ for an infinite G.P.]

Related Questions to study

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

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

parallel

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

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

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

parallel

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

$stack lim with n rightwards arrow infinity below invisible function application not stretchy sum from r equals 1 to n of fraction numerator r over denominator 1 cross times 3 cross times 5 cross times 7 cross times 9 cross times midline horizontal ellipsis cross times left parenthesis 2 r plus 1 right parenthesis end fraction blank$ is equal to

$stack lim with n rightwards arrow infinity below invisible function application not stretchy sum from r equals 1 to n of fraction numerator r over denominator 1 cross times 3 cross times 5 cross times 7 cross times 9 cross times midline horizontal ellipsis cross times left parenthesis 2 r plus 1 right parenthesis end fraction blank$ is equal to

If in a progression $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma$ etc., $left parenthesis a subscript r end subscript minus a subscript r plus 1 end subscript right parenthesis$ bears a constant ratio with $a subscript r end subscript cross times a subscript r plus 1 end subscript$ , then the terms of the progression are in

If in a progression $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma$ etc., $left parenthesis a subscript r end subscript minus a subscript r plus 1 end subscript right parenthesis$ bears a constant ratio with $a subscript r end subscript cross times a subscript r plus 1 end subscript$ , then the terms of the progression are in

parallel

If $a comma blank b$ , and $c$ are in G.P., then $a plus b comma blank 2 b$ , and $b plus c$ are in

If $a comma blank b$ , and $c$ are in G.P., then $a plus b comma blank 2 b$ , and $b plus c$ are in

If $a comma blank b comma$ and $c$ are in G.P., then the equations $a x to the power of 2 end exponent plus 2 b x plus c equals 0$ and $d x to the power of 2 end exponent plus 2 e x plus f equals 0$ have a common root if $fraction numerator d over denominator a end fraction comma fraction numerator e over denominator b end fraction comma fraction numerator f over denominator c end fraction$ are in

If $a comma blank b comma$ and $c$ are in G.P., then the equations $a x to the power of 2 end exponent plus 2 b x plus c equals 0$ and $d x to the power of 2 end exponent plus 2 e x plus f equals 0$ have a common root if $fraction numerator d over denominator a end fraction comma fraction numerator e over denominator b end fraction comma fraction numerator f over denominator c end fraction$ are in

Coefficients of $x to the power of 18 end exponent$ in $open parentheses 1 plus x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 18 x to the power of 18 end exponent close parentheses to the power of 2 end exponent$ is equal to

Coefficients of $x to the power of 18 end exponent$ in $open parentheses 1 plus x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 18 x to the power of 18 end exponent close parentheses to the power of 2 end exponent$ is equal to

parallel

If $a comma blank b comma$ and $c$ are in A.P., $p comma q$ and $r$ are in H.P. and $a p comma blank b q$ , and $c r$ are in G.P., then $fraction numerator p over denominator r end fraction plus fraction numerator r over denominator p end fraction$ is equal to

If $a comma blank b comma$ and $c$ are in A.P., $p comma q$ and $r$ are in H.P. and $a p comma blank b q$ , and $c r$ are in G.P., then $fraction numerator p over denominator r end fraction plus fraction numerator r over denominator p end fraction$ is equal to

If the sum of $m$ terms of an A.P. is the same as the sum of its $n$ terms, then the sum of its $left parenthesis m plus n right parenthesis$ terms is

If the sum of $m$ terms of an A.P. is the same as the sum of its $n$ terms, then the sum of its $left parenthesis m plus n right parenthesis$ terms is

The sum of 20 terms of a series of which every even term is 2 times the term before it, and every odd term is 3 times the term before it, the first term being unity is

The sum of 20 terms of a series of which every even term is 2 times the term before it, and every odd term is 3 times the term before it, the first term being unity is

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.