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Maths-

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Question

If $a comma fraction numerator 1 over denominator b end fraction comma blank c$ and $fraction numerator 1 over denominator p end fraction comma blank q comma fraction numerator 1 over denominator r end fraction$ form two arithmetic progressions of the same common difference, then $a comma blank q comma c$ are in A.P. if

$p, b, r$ are in A.P.
$\frac{1}{p}, \frac{1}{b}, \frac{1}{r}$ are in A.P.
$p, b, r$ are in G.P.
None of these

The correct answer is: $fraction numerator 1 over denominator p end fraction comma fraction numerator 1 over denominator b end fraction comma fraction numerator 1 over denominator r end fraction$ are in A.P.

Since $a comma blank q$ and $c$ are in A.P., so
$2 q equals a plus c$
$rightwards double arrow fraction numerator 1 over denominator p end fraction plus fraction numerator 1 over denominator r end fraction equals fraction numerator 2 over denominator b end fraction$
$rightwards double arrow fraction numerator 1 over denominator p end fraction comma fraction numerator 1 over denominator b end fraction comma fraction numerator 1 over denominator r end fraction$ are in A.P.

Related Questions to study

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

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

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

parallel

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 $fraction numerator 1 over denominator t subscript 1 end subscript end fraction plus fraction numerator 1 over denominator t subscript 2 end subscript end fraction plus fraction numerator 1 over denominator t subscript 3 end subscript end fraction plus horizontal ellipsis plus fraction numerator 1 over denominator t subscript 2003 end subscript end fraction equals$

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

parallel

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

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

parallel

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

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