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Maths-

General

Easy

Question

If the $m to the power of text th end text end exponent$ term of an $H P$ is $n$ and $n to the power of text th end text end exponent$ is $m$ , then the $r to the power of text th end text end exponent$ term will be

$\frac{r}{m n}$
$\frac{m n}{r + 1}$
$\frac{m n}{r}$
$\frac{m n}{r - 1}$

The correct answer is: $fraction numerator m n over denominator r end fraction$

$text Given end text T subscript m end subscript equals n comma T subscript n end subscript equals m text for HP. Therefore, for the corresponding end text$
$text AP end text m to the power of text th end text end exponent text term end text equals fraction numerator 1 over denominator n end fraction comma n to the power of text th end text end exponent text term end text equals fraction numerator 1 over denominator m end fraction text . end text$
Let a and d be the first term and common difference of this AP. Then
$table row cell a plus left parenthesis m minus 1 right parenthesis d equals fraction numerator 1 over denominator n end fraction end cell row cell a plus left parenthesis n minus 1 right parenthesis d equals fraction numerator 1 over denominator m end fraction end cell end table$
$text Solving these, we get end text a equals fraction numerator 1 over denominator m n end fraction comma d equals fraction numerator 1 over denominator m n end fraction$
$text Now, end text r to the power of text th end text end exponent text term of corresponding AP is end text$
$a plus left parenthesis r minus 1 right parenthesis d equals fraction numerator 1 over denominator m n end fraction plus left parenthesis r minus 1 right parenthesis fraction numerator 1 over denominator m n end fraction equals fraction numerator 1 plus r minus 1 over denominator m n end fraction equals fraction numerator r over denominator m n end fraction$
$text Therefore, end text r to the power of text th end text end exponent text term of corresponding end text H P text is end text fraction numerator m n over denominator r end fraction text . end text$
Note: Students should remember this question as a fact.

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Alcohols are ,acidic in nature because hydrogen is attached to oxygen which is an electronegative element. Alcohol is weaker acid than water because $O H to the power of minus end exponent$ ion is more stable than $R minus O to the power of minus end exponent$ ion $R minus 0 to the power of minus end exponent$ is less stable, due to +I effect of the alkyl group., Acidity of alcohols increases with increase:' in ' the stability of acid anion.

Some alcohols . like $C H subscript 3 end subscript minus O H comma C F subscript 3 end subscript minus C H subscript 2 end subscript minus O H$ , $blank open parentheses C F subscript 3 end subscript close parentheses subscript 3 end subscript C minus O H$
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