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Question

Statement 1:There are infinite geometric progressions for which 27, 8 and 12 are three of its terms (not necessarily consecutive)
Statement 2:Given terms are integers

Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
Statement 1 is True, Statement 2 is False
Statement 1 is False, Statement 2 is True

The correct answer is: Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1

Let, if possible, 8 be the first term and 12 and 27 be $n to the power of t h end exponent$ and $n to the power of t h end exponent$ terms, respectively. Then,
$12 equals a r to the power of m minus 1 end exponent equals 8 r to the power of m minus 1 end exponent comma blank 27 equals 8 r to the power of n minus 1 end exponent$
$rightwards double arrow fraction numerator 3 over denominator 2 end fraction equals r to the power of m minus 1 end exponent comma blank open parentheses fraction numerator 3 over denominator 2 end fraction close parentheses to the power of 3 end exponent equals r to the power of n minus 1 end exponent equals r to the power of 3 left parenthesis m minus 1 right parenthesis end exponent$
$rightwards double arrow n minus 1 equals 3 m minus 3$ or $3 m plus n plus 2$
$rightwards double arrow fraction numerator m over denominator 1 end fraction equals fraction numerator n plus 2 over denominator 3 end fraction equals k$ (say)
$therefore m equals k comma blank n equals 3 k minus 2$
By giving $k$ different values, we get the integral value of $m$ and $n$ . Hence there can be infinite number of G.P.’s whose any the three terms will be 8, 12, 27 (not consecutive). Obviously, statement 2 is not a correct explanation of statement 1

Related Questions to study

Statement 1:Sum of the series $1 to the power of 3 end exponent minus 2 to the power of 3 end exponent plus 3 to the power of 3 end exponent minus 4 to the power of 3 end exponent plus horizontal ellipsis plus 11 to the power of 3 end exponent equals 378$
Statement 2:For any odd integer $n greater or equal than 1 comma blank n to the power of 3 end exponent minus open parentheses n minus 1 close parentheses to the power of 3 end exponent plus horizontal ellipsis plus open parentheses negative 1 close parentheses to the power of n minus 1 end exponent blank 1 to the power of 3 end exponent equals fraction numerator 1 over denominator 4 end fraction open parentheses 2 n minus 1 close parentheses open parentheses n plus 1 close parentheses to the power of 2 end exponent$

Statement 1:Sum of the series $1 to the power of 3 end exponent minus 2 to the power of 3 end exponent plus 3 to the power of 3 end exponent minus 4 to the power of 3 end exponent plus horizontal ellipsis plus 11 to the power of 3 end exponent equals 378$
Statement 2:For any odd integer $n greater or equal than 1 comma blank n to the power of 3 end exponent minus open parentheses n minus 1 close parentheses to the power of 3 end exponent plus horizontal ellipsis plus open parentheses negative 1 close parentheses to the power of n minus 1 end exponent blank 1 to the power of 3 end exponent equals fraction numerator 1 over denominator 4 end fraction open parentheses 2 n minus 1 close parentheses open parentheses n plus 1 close parentheses to the power of 2 end exponent$

Statement 1:In a G.P. if the $open parentheses m plus n close parentheses to the power of t h end exponent$ term be $p$ and $open parentheses m minus n close parentheses to the power of t h end exponent$ term be $q$ , then its $m to the power of t h end exponent$ term is $square root of p q end root$
Statement 2: $T subscript m plus n end subscript comma blank T subscript m end subscript comma blank T subscript m minus n end subscript$ are in G.P.

Statement 1:In a G.P. if the $open parentheses m plus n close parentheses to the power of t h end exponent$ term be $p$ and $open parentheses m minus n close parentheses to the power of t h end exponent$ term be $q$ , then its $m to the power of t h end exponent$ term is $square root of p q end root$
Statement 2: $T subscript m plus n end subscript comma blank T subscript m end subscript comma blank T subscript m minus n end subscript$ are in G.P.

Statement 1:If $open vertical bar x minus 1 close vertical bar comma blank vertical line x minus 3 vertical line$ are first three terms of an AP, then its sixth term is 7< third terms.
Statement 2: $a comma blank a plus d comma blank a plus 2 d comma...$ are in AP $open parentheses d not equal to 0 close parentheses comma$ then sixth term is $open parentheses a plus 5 d close parentheses.$

Statement 1:If $open vertical bar x minus 1 close vertical bar comma blank vertical line x minus 3 vertical line$ are first three terms of an AP, then its sixth term is 7< third terms.
Statement 2: $a comma blank a plus d comma blank a plus 2 d comma...$ are in AP $open parentheses d not equal to 0 close parentheses comma$ then sixth term is $open parentheses a plus 5 d close parentheses.$

parallel

Statement 1:If sum f $n$ terms of a series $2 n to the power of 2 end exponent plus 3 n plus 1 comma$ then series is an AP.
Statement 2:Sum of $n$ terms of an AP is always of the form $p n to the power of 2 end exponent plus q n.$

Statement 1:If sum f $n$ terms of a series $2 n to the power of 2 end exponent plus 3 n plus 1 comma$ then series is an AP.
Statement 2:Sum of $n$ terms of an AP is always of the form $p n to the power of 2 end exponent plus q n.$

Let $a comma blank b comma blank c$ be three positive real numbers which are in HP.
Statement 1: $fraction numerator a plus b over denominator 2 a minus b end fraction plus fraction numerator c plus b over denominator 2 c minus b end fraction greater or equal than 4.$
Statement 2:If $x greater than 0 comma$ then $x plus fraction numerator 1 over denominator x end fraction greater or equal than 4.$

Let $a comma blank b comma blank c$ be three positive real numbers which are in HP.
Statement 1: $fraction numerator a plus b over denominator 2 a minus b end fraction plus fraction numerator c plus b over denominator 2 c minus b end fraction greater or equal than 4.$
Statement 2:If $x greater than 0 comma$ then $x plus fraction numerator 1 over denominator x end fraction greater or equal than 4.$

Statement 1:If $x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 25 z to the power of 2 end exponent equals x y z open parentheses fraction numerator 15 over denominator x end fraction plus fraction numerator 5 over denominator y end fraction plus fraction numerator 3 over denominator z end fraction close parentheses$ , then $x comma blank y comma blank z$ are in H.P.
Statement 2:If $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0$ , then $a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis a subscript n end subscript equals 0$

Statement 1:If $x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 25 z to the power of 2 end exponent equals x y z open parentheses fraction numerator 15 over denominator x end fraction plus fraction numerator 5 over denominator y end fraction plus fraction numerator 3 over denominator z end fraction close parentheses$ , then $x comma blank y comma blank z$ are in H.P.
Statement 2:If $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0$ , then $a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis a subscript n end subscript equals 0$

parallel

The harmonic mean of the roots of the equation $open parentheses 5 plus square root of 2 close parentheses x to the power of 2 end exponent minus open parentheses 4 plus square root of 5 close parentheses x plus 8 plus 2 square root of 5 equals 0$ is

The harmonic mean of the roots of the equation $open parentheses 5 plus square root of 2 close parentheses x to the power of 2 end exponent minus open parentheses 4 plus square root of 5 close parentheses x plus 8 plus 2 square root of 5 equals 0$ is

The sum to 50 terms of the series $fraction numerator 3 over denominator 1 to the power of 2 end exponent end fraction plus fraction numerator 5 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end fraction plus fraction numerator 7 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent plus 3 to the power of 2 end exponent end fraction plus horizontal ellipsis$ is

The sum to 50 terms of the series $fraction numerator 3 over denominator 1 to the power of 2 end exponent end fraction plus fraction numerator 5 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end fraction plus fraction numerator 7 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent plus 3 to the power of 2 end exponent end fraction plus horizontal ellipsis$ is

If $a comma blank x$ and $b$ are in A.P., $a comma blank y comma$ and $b$ are in G.P. and $a comma blank z comma blank b$ are in H.P. such that $x equals 9 z$ and $a greater than 0 comma blank b greater than 0$ , then

If $a comma blank x$ and $b$ are in A.P., $a comma blank y comma$ and $b$ are in G.P. and $a comma blank z comma blank b$ are in H.P. such that $x equals 9 z$ and $a greater than 0 comma blank b greater than 0$ , then

parallel

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma a subscript 10 end subscript$ be in A.P. and $h subscript 1 end subscript comma h subscript 2 end subscript comma horizontal ellipsis comma h subscript 10 end subscript$ be in H.P. If $a subscript 1 end subscript equals h subscript 1 end subscript equals 2$ and $a subscript 10 end subscript equals h subscript 10 end subscript equals 3$ , then $a subscript 4 end subscript h subscript 7 end subscript$ is

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma a subscript 10 end subscript$ be in A.P. and $h subscript 1 end subscript comma h subscript 2 end subscript comma horizontal ellipsis comma h subscript 10 end subscript$ be in H.P. If $a subscript 1 end subscript equals h subscript 1 end subscript equals 2$ and $a subscript 10 end subscript equals h subscript 10 end subscript equals 3$ , then $a subscript 4 end subscript h subscript 7 end subscript$ is

If $x comma blank y comma blank z$ are real and $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 16 z to the power of 2 end exponent minus 6 x y minus 12 y z minus 8 z x equals 0$ , then $x comma blank y comma blank z$ are in

If $x comma blank y comma blank z$ are real and $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 16 z to the power of 2 end exponent minus 6 x y minus 12 y z minus 8 z x equals 0$ , then $x comma blank y comma blank z$ are in

Let $alpha comma blank beta element of R$ . If $alpha comma blank beta to the power of 2 end exponent$ be the roots of quadratic equation $x to the power of 2 end exponent minus p x plus 1 equals 0$ and $alpha to the power of 2 end exponent comma beta$ be the roots of quadratic equation $x to the power of 2 end exponent minus q x plus 8 equals 0$ , then the value of ‘ $r$ ’ if $fraction numerator r over denominator 8 end fraction$ be arithmetic mean of $p$ and $q$ is

Let $alpha comma blank beta element of R$ . If $alpha comma blank beta to the power of 2 end exponent$ be the roots of quadratic equation $x to the power of 2 end exponent minus p x plus 1 equals 0$ and $alpha to the power of 2 end exponent comma beta$ be the roots of quadratic equation $x to the power of 2 end exponent minus q x plus 8 equals 0$ , then the value of ‘ $r$ ’ if $fraction numerator r over denominator 8 end fraction$ be arithmetic mean of $p$ and $q$ is

parallel

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

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

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

parallel

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