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

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma a subscript 4 end subscript$ and $a subscript 5 end subscript$ be such that $a subscript 1 end subscript comma a subscript 2 end subscript$ and $a subscript 3 end subscript$ are in A.P., $a subscript 2 end subscript comma a subscript 3 end subscript$ and $a subscript 4 end subscript$ are in G.P., and $a subscript 3 end subscript comma a subscript 4 end subscript$ and $a subscript 5 end subscript$ are in H.P. Then $log subscript e end subscript invisible function application a subscript 1 end subscript comma log subscript e end subscript invisible function application a subscript 3 end subscript$ and $log subscript e end subscript invisible function application a subscript 5 end subscript$ are in

G.P.
A.P.
H.P.
None of these

The correct answer is: A.P.

We have,
$a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript$ are in A.P. $rightwards double arrow 2 a subscript 2 end subscript equals a subscript 1 end subscript plus a subscript 3 end subscript$ (1)
$a subscript 2 end subscript comma a subscript 3 end subscript comma a subscript 4 end subscript$ are in G.P. $rightwards double arrow a subscript 3 end subscript superscript 2 end superscript equals a subscript 2 end subscript a subscript 4 end subscript$ (2)
$a subscript 3 end subscript comma a subscript 4 end subscript comma a subscript 5 end subscript$ are in H.P. $rightwards double arrow a subscript 4 end subscript equals fraction numerator 2 a subscript 3 end subscript a subscript 5 end subscript over denominator a subscript 3 end subscript plus a subscript 5 end subscript end fraction$ (3)
Putting $a subscript 2 end subscript equals fraction numerator a subscript 1 end subscript plus a subscript 3 end subscript over denominator 2 end fraction$ and $a subscript 4 end subscript equals fraction numerator 2 a subscript 3 end subscript a subscript 5 end subscript over denominator a subscript 3 end subscript plus a subscript 5 end subscript end fraction$ in (2), we get
$a subscript 3 end subscript superscript 2 end superscript equals fraction numerator a subscript 1 end subscript plus a subscript 3 end subscript over denominator 2 end fraction cross times fraction numerator 2 a subscript 3 end subscript a subscript 5 end subscript over denominator a subscript 3 end subscript plus a subscript 5 end subscript end fraction$
$rightwards double arrow a subscript 3 end subscript superscript 2 end superscript equals a subscript 1 end subscript a subscript 5 end subscript$
Hence, $a subscript 1 end subscript comma a subscript 3 end subscript$ , and $a subscript 5 end subscript$ are in G.P. So, $log subscript e end subscript invisible function application a subscript 1 end subscript comma log subscript e end subscript invisible function application a subscript 3 end subscript$ and $log subscript e end subscript invisible function application a subscript 5 end subscript$ are in A.P.

Related Questions to study

The $15 to the power of t h end exponent$ term of the series $2 fraction numerator 1 over denominator 2 end fraction plus 1 fraction numerator 7 over denominator 13 end fraction plus 1 fraction numerator 1 over denominator 9 end fraction plus fraction numerator 20 over denominator 23 end fraction plus horizontal ellipsis$ is

The $15 to the power of t h end exponent$ term of the series $2 fraction numerator 1 over denominator 2 end fraction plus 1 fraction numerator 7 over denominator 13 end fraction plus 1 fraction numerator 1 over denominator 9 end fraction plus fraction numerator 20 over denominator 23 end fraction plus horizontal ellipsis$ is

If $log subscript 2 end subscript invisible function application open parentheses 5 cross times 2 to the power of x end exponent plus 1 close parentheses comma log subscript 4 end subscript invisible function application left parenthesis 2 to the power of 1 minus x end exponent plus 1 right parenthesis$ and 1 are in A.P., then $x$ equals

If $log subscript 2 end subscript invisible function application open parentheses 5 cross times 2 to the power of x end exponent plus 1 close parentheses comma log subscript 4 end subscript invisible function application left parenthesis 2 to the power of 1 minus x end exponent plus 1 right parenthesis$ and 1 are in A.P., then $x$ equals

If $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis. comma a subscript n end subscript$ are in H.P., then
$fraction numerator a subscript 1 end subscript over denominator a subscript 2 end subscript plus a subscript 3 end subscript plus horizontal ellipsis plus a subscript n end subscript end fraction comma blank fraction numerator a subscript 2 end subscript over denominator a subscript 1 end subscript plus a subscript 3 end subscript plus horizontal ellipsis plus a subscript n end subscript end fraction comma horizontal ellipsis comma fraction numerator a subscript n end subscript over denominator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript n minus 1 end subscript end fraction$ are in

If $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis. comma a subscript n end subscript$ are in H.P., then
$fraction numerator a subscript 1 end subscript over denominator a subscript 2 end subscript plus a subscript 3 end subscript plus horizontal ellipsis plus a subscript n end subscript end fraction comma blank fraction numerator a subscript 2 end subscript over denominator a subscript 1 end subscript plus a subscript 3 end subscript plus horizontal ellipsis plus a subscript n end subscript end fraction comma horizontal ellipsis comma fraction numerator a subscript n end subscript over denominator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript n minus 1 end subscript end fraction$ are in

parallel

If $open parentheses 1 plus 3 plus 5 plus horizontal ellipsis plus p close parentheses plus open parentheses 1 plus 3 plus 5 plus horizontal ellipsis plus q close parentheses equals left parenthesis 1 plus 3 plus 5 plus horizontal ellipsis plus r right parenthesis$ where each set of parentheses contains the sum of of $p plus q plus r$ (where $p greater than 6$ ) is

If $open parentheses 1 plus 3 plus 5 plus horizontal ellipsis plus p close parentheses plus open parentheses 1 plus 3 plus 5 plus horizontal ellipsis plus q close parentheses equals left parenthesis 1 plus 3 plus 5 plus horizontal ellipsis plus r right parenthesis$ where each set of parentheses contains the sum of of $p plus q plus r$ (where $p greater than 6$ ) is

If $x comma blank y comma$ and $z$ are distinct prime numbers, then

If $x comma blank y comma$ and $z$ are distinct prime numbers, then

In a geometric series, the first term is $a$ and common ratio is $r$ . If $S subscript n end subscript$ denotes the sum of the $n$ terms and $U subscript n end subscript equals not stretchy sum from n equals 1 to n of S subscript n end subscript$ then $r S subscript n end subscript plus open parentheses 1 minus r close parentheses U subscript n end subscript$ equals

In a geometric series, the first term is $a$ and common ratio is $r$ . If $S subscript n end subscript$ denotes the sum of the $n$ terms and $U subscript n end subscript equals not stretchy sum from n equals 1 to n of S subscript n end subscript$ then $r S subscript n end subscript plus open parentheses 1 minus r close parentheses U subscript n end subscript$ equals

parallel

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis$ be terms of an A.P. If $fraction numerator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript p end subscript over denominator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript q end subscript end fraction equals fraction numerator p to the power of 2 end exponent over denominator q to the power of 2 end exponent end fraction comma blank p not equal to q$ , then $fraction numerator a subscript 6 end subscript over denominator a subscript 21 end subscript end fraction$ equals

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis$ be terms of an A.P. If $fraction numerator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript p end subscript over denominator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript q end subscript end fraction equals fraction numerator p to the power of 2 end exponent over denominator q to the power of 2 end exponent end fraction comma blank p not equal to q$ , then $fraction numerator a subscript 6 end subscript over denominator a subscript 21 end subscript end fraction$ equals

Consider the ten numbers $a r comma blank a r to the power of 2 end exponent comma a r to the power of 3 end exponent comma horizontal ellipsis comma a r to the power of 10 end exponent$ . If their sum is 18 and the sum of their reciprocals is 6 then the product of these ten numbers, is

Consider the ten numbers $a r comma blank a r to the power of 2 end exponent comma a r to the power of 3 end exponent comma horizontal ellipsis comma a r to the power of 10 end exponent$ . If their sum is 18 and the sum of their reciprocals is 6 then the product of these ten numbers, is

After striking the floor, a certain ball redounds $open parentheses 4 divided by 5 close parentheses to the power of t h end exponent$ of height from which it has fallen. Then the total distance that it travels before coming to rest, if it is gently dropped from a height of 120 m is

After striking the floor, a certain ball redounds $open parentheses 4 divided by 5 close parentheses to the power of t h end exponent$ of height from which it has fallen. Then the total distance that it travels before coming to rest, if it is gently dropped from a height of 120 m is

parallel

Greatest integer by which $1 plus not stretchy sum from r equals 1 to 30 of r cross times r factorial$ is divisible is

Greatest integer by which $1 plus not stretchy sum from r equals 1 to 30 of r cross times r factorial$ is divisible is

If $a comma blank b comma blank$ and $c$ are in A.P. and $p comma blank p to the power of ´ end exponent$ are, respectively, A.M. and G.M. between $a$ and $b$ while $q comma q to the power of ´ end exponent$ are, respectively, the A.M. and G.M. between $b$ and $c$ , then

If $a comma blank b comma blank$ and $c$ are in A.P. and $p comma blank p to the power of ´ end exponent$ are, respectively, A.M. and G.M. between $a$ and $b$ while $q comma q to the power of ´ end exponent$ are, respectively, the A.M. and G.M. between $b$ and $c$ , then

If the ratio of the sum to $n$ terms of two A.P.’s is $open parentheses 5 n plus 3 close parentheses colon left parenthesis 3 n plus 4 right parenthesis$ , then the ratio of their 17^th terms is

If the ratio of the sum to $n$ terms of two A.P.’s is $open parentheses 5 n plus 3 close parentheses colon left parenthesis 3 n plus 4 right parenthesis$ , then the ratio of their 17^th terms is

parallel

If the $p to the power of t h end exponent comma blank q to the power of t h end exponent comma blank r to the power of t h end exponent$ and $s to the power of t h end exponent$ terms of an A.P. are in G.P., then $p minus q comma blank q minus r comma blank r minus s$ are in

If the $p to the power of t h end exponent comma blank q to the power of t h end exponent comma blank r to the power of t h end exponent$ and $s to the power of t h end exponent$ terms of an A.P. are in G.P., then $p minus q comma blank q minus r comma blank r minus s$ are in

Let $alpha$ , and $beta$ be the roots of $x to the power of 2 end exponent minus x plus p equals 0$ and $gamma$ and $delta$ be the root of $x to the power of 2 end exponent minus 4 x plus q equals 0$ . If $alpha comma blank beta comma blank a n d blank gamma comma blank delta$ are in G.P., then the integral values of $p$ and $q$ , respectively, are

Let $alpha$ , and $beta$ be the roots of $x to the power of 2 end exponent minus x plus p equals 0$ and $gamma$ and $delta$ be the root of $x to the power of 2 end exponent minus 4 x plus q equals 0$ . If $alpha comma blank beta comma blank a n d blank gamma comma blank delta$ are in G.P., then the integral values of $p$ and $q$ , respectively, are

If $x comma blank y comma$ and $z$ are in G.P., and $x plus 3 comma blank y plus 3$ , and $z plus 3$ are in H.P., then

If $x comma blank y comma$ and $z$ are in G.P., and $x plus 3 comma blank y plus 3$ , and $z plus 3$ are in H.P., 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.