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Question

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

$\frac{n 2^{n} - 1}{n + 1}$
$\frac{n + 1}{2^{n + 1} (n + 2)}$
$\frac{n}{2^{n}} - 1$
$\frac{(n + 1) 2^{n} - 1}{2^{n + 1} (n + 1)}$

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

$t subscript r end subscript equals fraction numerator 1 over denominator r times 2 to the power of r end exponent end fraction minus fraction numerator 1 over denominator left parenthesis r plus 1 right parenthesis 2 to the power of p plus 1 end exponent end fraction$
$rightwards double arrow S subscript n end subscript equals fraction numerator 1 over denominator 2 end fraction minus fraction numerator 1 over denominator 2 to the power of n plus 1 end exponent left parenthesis n plus 1 right parenthesis end fraction equals fraction numerator left parenthesis n plus 1 right parenthesis 2 to the power of n end exponent minus 1 over denominator 2 to the power of n plus 1 end exponent left parenthesis n plus 1 right parenthesis end fraction$

Related Questions to study

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

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

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

How many isomers of C₅H₁₁OH will be primary alcohols?

How many isomers of C₅H₁₁OH will be primary alcohols?

Chemistry-General

The number of solutions the equation $x to the power of 3 end exponent plus 2 x to the power of 2 end exponent plus 5 x plus 2 c o s invisible function application x equals 0 text in end text left square bracket 0 , 2 pi right square bracket$ is

The number of solutions the equation $x to the power of 3 end exponent plus 2 x to the power of 2 end exponent plus 5 x plus 2 c o s invisible function application x equals 0 text in end text left square bracket 0 , 2 pi right square bracket$ is

$text If end text theta element of left square bracket 0 , 5 pi right square bracket text and end text r element of R text such that end text 2 s i n invisible function application theta equals r to the power of 4 end exponent minus 2 r to the power of 2 end exponent plus 3$ then the maximum number of values of the pair $left parenthesis r comma theta right parenthesis$ is

$text If end text theta element of left square bracket 0 , 5 pi right square bracket text and end text r element of R text such that end text 2 s i n invisible function application theta equals r to the power of 4 end exponent minus 2 r to the power of 2 end exponent plus 3$ then the maximum number of values of the pair $left parenthesis r comma theta right parenthesis$ is

parallel

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