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Maths-

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Question

In a sequence of (4n + 1) terms the first (2n + 1) terms are in AP whose common difference is 2 and the last (2n + 1) terms are in GP whose common ratio is 0.5. If the middle terms of the AP and GP are equal then the middle term of the sequence is

$\frac{n \cdot 2^{n + 1}}{2^{n} - 1}$
$\frac{n \cdot 2^{n + 1}}{2^{2 n} - 1}$
$n \cdot 2^{n}$
$None of these$

The correct answer is: $fraction numerator n times 2 to the power of n plus 1 end exponent over denominator 2 to the power of n end exponent minus 1 end fraction$

$text Let the sequence is end text a comma a plus 2 comma a plus 4 comma horizontal ellipsis comma a plus 4 n text , end text$
$fraction numerator a plus 4 n over denominator 2 end fraction comma fraction numerator a plus 4 n over denominator 4 end fraction comma midline horizontal ellipsis comma fraction numerator a plus 4 n over denominator 2 to the power of 2 n end exponent end fraction$
It is given that
$a plus 2 n equals fraction numerator a plus 4 n over denominator 2 to the power of n end exponent end fraction$
$rightwards double arrow a open parentheses 1 minus fraction numerator 1 over denominator 2 to the power of n end exponent end fraction close parentheses equals fraction numerator 4 n over denominator 2 to the power of n end exponent end fraction minus 2 n rightwards double arrow a equals fraction numerator 4 n minus 2 n times 2 to the power of n end exponent over denominator 2 to the power of n end exponent minus 1 end fraction$
$text Therefore, middle term is end text a plus 4 n equals fraction numerator 2 n minus open parentheses 2 minus 2 to the power of n end exponent close parentheses over denominator 2 to the power of n end exponent minus 1 end fraction plus 4 n equals fraction numerator 2 n times 2 to the power of n end exponent over denominator 2 to the power of n end exponent minus 1 end fraction$

Related Questions to study

If $x subscript i end subscript greater than 0 comma i equals 1 , 2 comma horizontal ellipsis comma 50$ and $x subscript 1 end subscript plus x subscript 2 end subscript plus midline horizontal ellipsis plus x subscript 50 end subscript equals 50$ , then the minimum value of $fraction numerator 1 over denominator x subscript 1 end subscript end fraction plus fraction numerator 1 over denominator x subscript 2 end subscript end fraction plus midline horizontal ellipsis plus fraction numerator 1 over denominator x subscript 50 end subscript end fraction$ equals

If $x subscript i end subscript greater than 0 comma i equals 1 , 2 comma horizontal ellipsis comma 50$ and $x subscript 1 end subscript plus x subscript 2 end subscript plus midline horizontal ellipsis plus x subscript 50 end subscript equals 50$ , then the minimum value of $fraction numerator 1 over denominator x subscript 1 end subscript end fraction plus fraction numerator 1 over denominator x subscript 2 end subscript end fraction plus midline horizontal ellipsis plus fraction numerator 1 over denominator x subscript 50 end subscript end fraction$ equals

If $fraction numerator 3 plus 5 plus 7 plus midline horizontal ellipsis plus n text terms end text over denominator 5 plus 8 plus 11 plus midline horizontal ellipsis plus 10 text terms end text end fraction equals 7$ , then the value of $n$ is

If $fraction numerator 3 plus 5 plus 7 plus midline horizontal ellipsis plus n text terms end text over denominator 5 plus 8 plus 11 plus midline horizontal ellipsis plus 10 text terms end text end fraction equals 7$ , then the value of $n$ is

If $t subscript r end subscript equals fraction numerator r plus 2 over denominator r left parenthesis r plus 1 right parenthesis end fraction cross times fraction numerator 1 over denominator 2 to the power of r plus 1 end exponent end fraction$ , then $sum subscript r equals 1 end subscript superscript n end superscript t subscript r end subscript$ is equal to

If $t subscript r end subscript equals fraction numerator r plus 2 over denominator r left parenthesis r plus 1 right parenthesis end fraction cross times fraction numerator 1 over denominator 2 to the power of r plus 1 end exponent end fraction$ , then $sum subscript r equals 1 end subscript superscript n end superscript t subscript r end subscript$ is equal to

parallel

The sum of $fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 times 2 to the power of 3 end exponent end fraction plus fraction numerator 1 over denominator 5 times 2 to the power of 5 end exponent end fraction plus midline horizontal ellipsis infinity$ is

The sum of $fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 times 2 to the power of 3 end exponent end fraction plus fraction numerator 1 over denominator 5 times 2 to the power of 5 end exponent end fraction plus midline horizontal ellipsis infinity$ is

$1 plus fraction numerator 1 plus 2 over denominator 2 factorial end fraction plus fraction numerator 1 plus 2 plus 3 over denominator 3 factorial end fraction plus fraction numerator 1 plus 2 plus 3 plus 4 over denominator 4 factorial end fraction plus midline horizontal ellipsis infinity equals$

$1 plus fraction numerator 1 plus 2 over denominator 2 factorial end fraction plus fraction numerator 1 plus 2 plus 3 over denominator 3 factorial end fraction plus fraction numerator 1 plus 2 plus 3 plus 4 over denominator 4 factorial end fraction plus midline horizontal ellipsis infinity equals$

The sum of the series $1 times 3 to the power of 2 end exponent plus 2 times 5 to the power of 2 end exponent plus 3 times 7 to the power of 2 end exponent plus midline horizontal ellipsis$ up to 20 terms is

The sum of the series $1 times 3 to the power of 2 end exponent plus 2 times 5 to the power of 2 end exponent plus 3 times 7 to the power of 2 end exponent plus midline horizontal ellipsis$ up to 20 terms is

parallel

$text If the end text left parenthesis m plus 1 right parenthesis to the power of text th end text end exponent comma left parenthesis n plus 1 right parenthesis to the power of text th end text end exponent text and end text left parenthesis r plus 1 right parenthesis to the power of text th end text end exponent text terms of an AP are in GP end text$ 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

$text If the end text left parenthesis m plus 1 right parenthesis to the power of text th end text end exponent comma left parenthesis n plus 1 right parenthesis to the power of text th end text end exponent text and end text left parenthesis r plus 1 right parenthesis to the power of text th end text end exponent text terms of an AP are in GP end text$ 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

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

parallel

If a ,b, c are three distinct positive real numbers which are in $text HP, then end text fraction numerator 3 a plus 2 b over denominator 2 a minus b end fraction plus fraction numerator 3 c plus 2 b over denominator 2 c minus b end fraction text is end text$

If a ,b, c are three distinct positive real numbers which are in $text HP, then end text fraction numerator 3 a plus 2 b over denominator 2 a minus b end fraction plus fraction numerator 3 c plus 2 b over denominator 2 c minus b end fraction text is end text$

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

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

If $left parenthesis a plus 2 b plus 2 c right parenthesis left parenthesis a minus 2 b plus 2 c right parenthesis equals a to the power of 2 end exponent plus 4 c to the power of 2 end exponent text , end text$ where a ,b, c are nonzero numbers. Then a ,b,c are in

If $left parenthesis a plus 2 b plus 2 c right parenthesis left parenthesis a minus 2 b plus 2 c right parenthesis equals a to the power of 2 end exponent plus 4 c to the power of 2 end exponent text , end text$ where a ,b, c are nonzero numbers. Then a ,b,c are in

parallel

The sum of infinite terms of a GP is x and on squaring the each term of it, the sum will be y. Then the common ratio of this series is

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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 $C H subscript 3 end subscript minus O H comma C F subscript 3 end subscript minus C H subscript 2 end subscript minus O H$ , $blank open parentheses C F subscript 3 end subscript close parentheses subscript 3 end subscript C minus O H$
are more acidic than water:
The strongest conjugate base out of the following is :

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 $C H subscript 3 end subscript minus O H comma C F subscript 3 end subscript minus C H subscript 2 end subscript minus O H$ , $blank open parentheses C F subscript 3 end subscript close parentheses subscript 3 end subscript C minus O H$
are more acidic than water:
The strongest conjugate base out of the following is :

Chemistry-General

Final product (D) in the sequence of reactions is:

Final product (D) in the sequence of reactions is:

Chemistry-General

parallel

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