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Question

Statement 1: $1 to the power of 99 end exponent plus 2 to the power of 99 end exponent plus horizontal ellipsis plus 100 to the power of 99 end exponent$ is divisible by 10100
Statement 2: $a to the power of n end exponent plus b to the power of n end exponent$ is divisible by $a plus b$ if $n$ is odd

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 correct explanation for Statement 1

Statement 2 is true as
$fraction numerator a to the power of n end exponent plus b to the power of n end exponent over denominator a plus b end fraction equals fraction numerator a to the power of n end exponent minus open parentheses negative b close parentheses to the power of n end exponent over denominator a minus left parenthesis negative b right parenthesis end fraction equals a to the power of n minus 1 end exponent minus a to the power of n minus 2 end exponent b plus a to the power of n minus 3 end exponent b to the power of 2 end exponent minus horizontal ellipsis open parentheses negative 1 close parentheses to the power of n minus 1 end exponent b to the power of n minus 1 end exponent$
Now,
$1 to the power of 99 end exponent plus 2 to the power of 99 end exponent plus horizontal ellipsis 100 to the power of 99 end exponent equals open parentheses 1 to the power of 99 end exponent plus 100 to the power of 99 end exponent close parentheses plus open parentheses 2 to the power of 99 end exponent plus 99 to the power of 99 end exponent close parentheses plus horizontal ellipsis plus open parentheses 50 to the power of 99 end exponent plus 51 to the power of 99 end exponent close parentheses$
Each bracket is divisible by 101; hence the sum is divided by 101. Also,
$1 to the power of 99 end exponent plus 2 to the power of 99 end exponent plus L horizontal ellipsis 100 to the power of 99 end exponent equals open parentheses 1 to the power of 99 end exponent plus 99 to the power of 99 end exponent close parentheses plus open parentheses 2 to the power of 99 end exponent plus 98 to the power of 99 end exponent close parentheses$
$plus horizontal ellipsis plus open parentheses 49 to the power of 99 end exponent plus 51 to the power of 99 end exponent close parentheses plus 50 to the power of 99 end exponent plus 100 to the power of 99 end exponent$
Here, each bracket and $50 to the power of 99 end exponent$ and $100 to the power of 99 end exponent$ are divisible by 100. Hence sum is divisible by 100. Hence sum is divisible by $101 cross times 100 equals 10100$

Related Questions to study

Statement 1:Let $F subscript 1 end subscript open parentheses n close parentheses equals 1 plus fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 end fraction plus... plus fraction numerator 1 over denominator n end fraction$ , then $not stretchy sum from r equals 1 to n of F subscript 1 end subscript open parentheses r close parentheses equals open parentheses n plus 1 close parentheses F subscript 1 end subscript open parentheses n close parentheses minus n.$
Statement 2: $fraction numerator 1 to the power of negative 1 end exponent plus 2 to the power of negative 1 end exponent plus 3 to the power of negative 1 end exponent plus... plus n to the power of negative 1 end exponent over denominator n end fraction greater than open parentheses fraction numerator 1 plus 2 plus 3 plus... plus n over denominator n end fraction close parentheses to the power of negative 1 end exponent$ or $open parentheses 1 plus fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 end fraction plus plus... fraction numerator 1 over denominator n end fraction close parentheses greater than fraction numerator n to the power of 2 end exponent over denominator not stretchy sum n end fraction$ or $open parentheses 1 plus fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 end fraction plus... plus fraction numerator 1 over denominator n end fraction close parentheses greater than fraction numerator 2 n over denominator left parenthesis n plus 1 right parenthesis end fraction$

Statement 1:Let $F subscript 1 end subscript open parentheses n close parentheses equals 1 plus fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 end fraction plus... plus fraction numerator 1 over denominator n end fraction$ , then $not stretchy sum from r equals 1 to n of F subscript 1 end subscript open parentheses r close parentheses equals open parentheses n plus 1 close parentheses F subscript 1 end subscript open parentheses n close parentheses minus n.$
Statement 2: $fraction numerator 1 to the power of negative 1 end exponent plus 2 to the power of negative 1 end exponent plus 3 to the power of negative 1 end exponent plus... plus n to the power of negative 1 end exponent over denominator n end fraction greater than open parentheses fraction numerator 1 plus 2 plus 3 plus... plus n over denominator n end fraction close parentheses to the power of negative 1 end exponent$ or $open parentheses 1 plus fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 end fraction plus plus... fraction numerator 1 over denominator n end fraction close parentheses greater than fraction numerator n to the power of 2 end exponent over denominator not stretchy sum n end fraction$ or $open parentheses 1 plus fraction numerator 1 over denominator 2 end fraction plus fraction numerator 1 over denominator 3 end fraction plus... plus fraction numerator 1 over denominator n end fraction close parentheses greater than fraction numerator 2 n over denominator left parenthesis n plus 1 right parenthesis end fraction$

Statement 1:If an infinite G.P. has 2^nd term $x$ and its sum is 4, then $x$ belongs to $left parenthesis negative 8 comma blank 1 right parenthesis$
Statement 2:Sum of an infinite G.P. is finite if for its common ratio $r comma blank 0 less than open vertical bar r close vertical bar less than 1$

Statement 1:If an infinite G.P. has 2^nd term $x$ and its sum is 4, then $x$ belongs to $left parenthesis negative 8 comma blank 1 right parenthesis$
Statement 2:Sum of an infinite G.P. is finite if for its common ratio $r comma blank 0 less than open vertical bar r close vertical bar less than 1$

Statement 1:Coefficient of $x to the power of 14 end exponent$ in $open parentheses 1 plus 2 x plus 3 x to the power of 2 end exponent plus horizontal ellipsis plus 16 x to the power of 15 end exponent close parentheses to the power of 2 end exponent$ is 560
Statement 2: $not stretchy sum from r equals 1 to n of r left parenthesis n minus r right parenthesis equals fraction numerator n left parenthesis n to the power of 2 end exponent minus 1 right parenthesis over denominator 6 end fraction$

Statement 1:Coefficient of $x to the power of 14 end exponent$ in $open parentheses 1 plus 2 x plus 3 x to the power of 2 end exponent plus horizontal ellipsis plus 16 x to the power of 15 end exponent close parentheses to the power of 2 end exponent$ is 560
Statement 2: $not stretchy sum from r equals 1 to n of r left parenthesis n minus r right parenthesis equals fraction numerator n left parenthesis n to the power of 2 end exponent minus 1 right parenthesis over denominator 6 end fraction$

parallel

Statement 1: $x equals 1111 midline horizontal ellipsis 91$ times is composite number
Statement 2:91 is composite number

Statement 1: $x equals 1111 midline horizontal ellipsis 91$ times is composite number
Statement 2:91 is composite number

Statement 1:The numbers $square root of 2 comma blank square root of 3 comma blank square root of 5$ cannot be the terms of a single A.P. with non-zero common difference
Statement 2:If $p comma blank q comma blank r left parenthesis p not equal to q right parenthesis$ are terms (not necessarily consecutive) of an A.P., then there exists a rational number $k$ such that $left parenthesis r minus q right parenthesis divided by left parenthesis q minus p right parenthesis equals k$

Statement 1:The numbers $square root of 2 comma blank square root of 3 comma blank square root of 5$ cannot be the terms of a single A.P. with non-zero common difference
Statement 2:If $p comma blank q comma blank r left parenthesis p not equal to q right parenthesis$ are terms (not necessarily consecutive) of an A.P., then there exists a rational number $k$ such that $left parenthesis r minus q right parenthesis divided by left parenthesis q minus p right parenthesis equals k$

Statement 1:3, 6, 12 are in GP, then 9, 12, 18 are in HP.
Statement 2:If middle term is added in three consecutive terms of a GP, resultant will be in HP.

Statement 1:3, 6, 12 are in GP, then 9, 12, 18 are in HP.
Statement 2:If middle term is added in three consecutive terms of a GP, resultant will be in HP.

parallel

Statement 1:If sum of $n$ terms of a series is $6 n to the power of 2 end exponent plus 3 n plus 1$ then the series is in AP $.$
Statement 2:Sum of $n$ terms of an AP is always of the form $a n to the power of 2 end exponent plus b n.$

Statement 1:If sum of $n$ terms of a series is $6 n to the power of 2 end exponent plus 3 n plus 1$ then the series is in AP $.$
Statement 2:Sum of $n$ terms of an AP is always of the form $a n to the power of 2 end exponent plus b n.$

Statement 1:If the arithmetic mean of two numbers is 5/2, geometric mean of the numbers is 2, then the harmonic mean will be 8/5
Statement 2:For a group of positive numbers $open parentheses G. M. close parentheses to the power of 2 end exponent equals open parentheses A. M. close parentheses cross times left parenthesis H. M. right parenthesis$

Statement 1:If the arithmetic mean of two numbers is 5/2, geometric mean of the numbers is 2, then the harmonic mean will be 8/5
Statement 2:For a group of positive numbers $open parentheses G. M. close parentheses to the power of 2 end exponent equals open parentheses A. M. close parentheses cross times left parenthesis H. M. right parenthesis$

Statement 1:The sum of $n$ terms of two arithmetic progressions are in the ratio $open parentheses 7 n plus 1 close parentheses colon open parentheses 4 n plus 17 close parentheses comma$ then the ratio of their $n$ th terms is 7: 4.
Statement 2:If $S subscript n end subscript equals a x to the power of 2 end exponent plus b x plus c comma$ then $T subscript n end subscript equals S subscript n end subscript minus S subscript n minus 1 end subscript$

Statement 1:The sum of $n$ terms of two arithmetic progressions are in the ratio $open parentheses 7 n plus 1 close parentheses colon open parentheses 4 n plus 17 close parentheses comma$ then the ratio of their $n$ th terms is 7: 4.
Statement 2:If $S subscript n end subscript equals a x to the power of 2 end exponent plus b x plus c comma$ then $T subscript n end subscript equals S subscript n end subscript minus S subscript n minus 1 end subscript$

parallel

Let $a comma blank r element of R minus left curly bracket 0 comma blank 1 comma blank minus 1 right curly bracket$ and $n$ be an even number
Statement 1: $a cross times a r cross times a r to the power of 2 end exponent midline horizontal ellipsis a r to the power of n minus 1 end exponent equals open parentheses a to the power of 2 end exponent r to the power of n minus 1 end exponent close parentheses to the power of n divided by 2 end exponent$
Statement 2:Product of $i to the power of t h end exponent$ term from the beginning and from the end in a G.P. is independent of $i$

Let $a comma blank r element of R minus left curly bracket 0 comma blank 1 comma blank minus 1 right curly bracket$ and $n$ be an even number
Statement 1: $a cross times a r cross times a r to the power of 2 end exponent midline horizontal ellipsis a r to the power of n minus 1 end exponent equals open parentheses a to the power of 2 end exponent r to the power of n minus 1 end exponent close parentheses to the power of n divided by 2 end exponent$
Statement 2:Product of $i to the power of t h end exponent$ term from the beginning and from the end in a G.P. is independent of $i$

Statement 1:Let $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ and $x$ be distinct real number such that $open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript superscript 2 end superscript close parentheses x to the power of 2 end exponent plus 2 open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript blank p subscript r plus 1 end subscript close parentheses x plus not stretchy sum from r equals 2 to n of p subscript r end subscript superscript 2 end superscript less or equal than 0$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P. and when $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus a subscript 3 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0 comma blank a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis equals a subscript n end subscript equals 0$
Statement 2:If $fraction numerator p subscript 2 end subscript over denominator p subscript 1 end subscript end fraction equals fraction numerator p subscript 3 end subscript over denominator p subscript 2 end subscript end fraction equals horizontal ellipsis equals fraction numerator p subscript n end subscript over denominator p subscript n minus 1 end subscript end fraction$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P.

Statement 1:Let $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ and $x$ be distinct real number such that $open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript superscript 2 end superscript close parentheses x to the power of 2 end exponent plus 2 open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript blank p subscript r plus 1 end subscript close parentheses x plus not stretchy sum from r equals 2 to n of p subscript r end subscript superscript 2 end superscript less or equal than 0$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P. and when $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus a subscript 3 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0 comma blank a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis equals a subscript n end subscript equals 0$
Statement 2:If $fraction numerator p subscript 2 end subscript over denominator p subscript 1 end subscript end fraction equals fraction numerator p subscript 3 end subscript over denominator p subscript 2 end subscript end fraction equals horizontal ellipsis equals fraction numerator p subscript n end subscript over denominator p subscript n minus 1 end subscript end fraction$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P.

Statement 1:If $3 x plus 4 y equals 5 comma$ then the greatest value of $x to the power of 2 end exponent y to the power of 3 end exponent$ is $fraction numerator 3 over denominator 16 end fraction.$
Statement 2:Greatest value occurs when $9 x equals 8 y.$

Statement 1:If $3 x plus 4 y equals 5 comma$ then the greatest value of $x to the power of 2 end exponent y to the power of 3 end exponent$ is $fraction numerator 3 over denominator 16 end fraction.$
Statement 2:Greatest value occurs when $9 x equals 8 y.$

parallel

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: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: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.

parallel

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