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Maths-

General

Easy

Question

$text If the end text left parenthesis m plus 1 right parenthesis to the power of text th end text end exponent comma left parenthesis n plus 1 right parenthesis to the power of text th end text end exponent text and end text left parenthesis r plus 1 right parenthesis to the power of text th end text end exponent text terms of an AP are in GP end text$ and m, n, r are in HP, then the value of the ratio of the common difference to the first term of the AP is

$- \frac{2}{n}$
$\frac{2}{n}$
$- \frac{n}{2}$
$\frac{n}{2}$

The correct answer is: $negative fraction numerator 2 over denominator n end fraction$

Let a be the first term and d be the common difference of the given AP. Then as given the $left parenthesis m plus 1 right parenthesis to the power of text th end text end exponent comma left parenthesis n plus 1 right parenthesis to the power of text th end text end exponent text and end text left parenthesis r plus 1 right parenthesis to the power of text th end text end exponent$ terms are in GP. So, $blank a plus m d comma a plus n d comma a plus r d text are in GP. This end text$
gives
$table row cell left parenthesis a plus m d right parenthesis to the power of 2 end exponent equals left parenthesis a plus m d right parenthesis left parenthesis a plus r d right parenthesis end cell row cell rightwards double arrow a left parenthesis 2 n minus m minus r right parenthesis equals d open parentheses m r minus n to the power of 2 end exponent close parentheses end cell row cell rightwards double arrow fraction numerator d over denominator a end fraction equals fraction numerator 2 n minus left parenthesis m plus r right parenthesis over denominator m r minus n to the power of 2 end exponent end fraction end cell end table$
$text Next, end text m comma n comma r text in HP. So end text$
$n equals fraction numerator 2 m r over denominator m plus r end fraction$
From Eqs. (1) and (2)
$fraction numerator d over denominator a end fraction minus fraction numerator 2 n minus left parenthesis m plus r right parenthesis over denominator m r minus n to the power of 2 end exponent end fraction minus fraction numerator 2 over denominator n end fraction open parentheses fraction numerator 2 n minus left parenthesis m plus r right parenthesis over denominator left parenthesis m plus r right parenthesis minus 2 n end fraction close parentheses minus negative fraction numerator 2 over denominator n end fraction$

Related Questions to study

If the harmonic mean between a and b is H, then $fraction numerator H plus a over denominator H minus a end fraction plus fraction numerator H plus b over denominator H minus b end fraction equals$

If the harmonic mean between a and b is H, then $fraction numerator H plus a over denominator H minus a end fraction plus fraction numerator H plus b over denominator H minus b end fraction equals$

If $fraction numerator a to the power of n plus 1 end exponent plus b to the power of n plus 1 end exponent over denominator a to the power of n end exponent plus b to the power of n end exponent end fraction$ is the harmonic mean between $a$ and $b$ , then the value of $n$ is

If $fraction numerator a to the power of n plus 1 end exponent plus b to the power of n plus 1 end exponent over denominator a to the power of n end exponent plus b to the power of n end exponent end fraction$ is the harmonic mean between $a$ and $b$ , then the value of $n$ is

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parallel

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are more acidic than water:
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Chemistry-General

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Chemistry-General

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parallel

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Which of the following will be the major product when

is treated with dil. H₂SO₄ in the presence of HgSO₄ ?

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is treated with dil. H₂SO₄ in the presence of HgSO₄ ?

Chemistry-General

parallel

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Chemistry-General

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Chemistry-General

In the following reaction,

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Chemistry-General

parallel

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