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Maths-

General

Easy

Question

If the sum of $n$ terms of an A.P. is $c n left parenthesis n minus 1 right parenthesis$ , where $c not equal to 0$ , then sum of the squares of these terms is

$c^{2} n {(n + 1)}^{2}$
$\frac{2}{3} c^{2} n (n - 1) (2 n - 1)$
$\frac{2 c^{2}}{3} n (n + 1) (2 n + 1)$
None of these

The correct answer is: $fraction numerator 2 over denominator 3 end fraction c to the power of 2 end exponent n open parentheses n minus 1 close parentheses left parenthesis 2 n minus 1 right parenthesis$

If $t subscript r end subscript$ be the $r to the power of t h end exponent$ term of the A.P., then
$t subscript r end subscript equals S subscript r end subscript minus S subscript r minus 1 end subscript$
$equals c r blank open parentheses r minus 1 close parentheses minus c open parentheses r minus 1 close parentheses open parentheses r minus 2 close parentheses$
$equals c open parentheses r minus 1 close parentheses open parentheses r minus r plus 2 close parentheses equals 2 c open parentheses r minus 1 close parentheses$
We have,
$t subscript 1 end subscript superscript 2 end superscript plus t subscript 2 end subscript superscript 2 end superscript plus horizontal ellipsis plus t subscript n end subscript superscript 2 end superscript equals 4 c to the power of 2 end exponent left parenthesis 0 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent plus horizontal ellipsis plus open parentheses n minus 1 close parentheses to the power of 2 end exponent right parenthesis$
$equals 4 c to the power of 2 end exponent fraction numerator open parentheses n minus 1 close parentheses n left parenthesis 2 n minus 1 right parenthesis over denominator 6 end fraction$
$equals fraction numerator 2 over denominator 3 end fraction blank c to the power of 2 end exponent n blank left parenthesis n minus 1 right parenthesis left parenthesis 2 n minus 1 right parenthesis$

Related Questions to study

Find the sum $open parentheses x plus 2 close parentheses to the power of n minus 1 end exponent plus open parentheses x plus 2 close parentheses to the power of n minus 2 end exponent open parentheses x plus 1 close parentheses plus open parentheses x plus 2 close parentheses to the power of n minus 3 end exponent open parentheses x plus 1 close parentheses to the power of 2 end exponent plus horizontal ellipsis plus open parentheses x plus 1 close parentheses to the power of n minus 1 end exponent$

Find the sum $open parentheses x plus 2 close parentheses to the power of n minus 1 end exponent plus open parentheses x plus 2 close parentheses to the power of n minus 2 end exponent open parentheses x plus 1 close parentheses plus open parentheses x plus 2 close parentheses to the power of n minus 3 end exponent open parentheses x plus 1 close parentheses to the power of 2 end exponent plus horizontal ellipsis plus open parentheses x plus 1 close parentheses to the power of n minus 1 end exponent$

If $S subscript p end subscript$ denotes the sum of the series $1 minus r to the power of p end exponent plus r to the power of 2 p end exponent minus r to the power of 3 p end exponent plus midline horizontal ellipsis$ to $infinity$ and $S subscript p end subscript$ the sum of the series $1 minus r to the power of p end exponent plus r to the power of 2 p end exponent minus r to the power of 3 p end exponent plus horizontal ellipsis$ to $infinity comma blank open vertical bar r close vertical bar less than 1$ , then $S subscript p end subscript plus s subscript p end subscript$ in terms of $S subscript 2 p end subscript$ is

If $S subscript p end subscript$ denotes the sum of the series $1 minus r to the power of p end exponent plus r to the power of 2 p end exponent minus r to the power of 3 p end exponent plus midline horizontal ellipsis$ to $infinity$ and $S subscript p end subscript$ the sum of the series $1 minus r to the power of p end exponent plus r to the power of 2 p end exponent minus r to the power of 3 p end exponent plus horizontal ellipsis$ to $infinity comma blank open vertical bar r close vertical bar less than 1$ , then $S subscript p end subscript plus s subscript p end subscript$ in terms of $S subscript 2 p end subscript$ is

If $R$ is a relation on a finite set having $n$ elements, then the number of relations on $A$ is

If $R$ is a relation on a finite set having $n$ elements, then the number of relations on $A$ is

parallel

A relation between two persons is defined as follows:
$a R b left right double arrow a$ and $b$ born in different months. Then, $R$ is

A relation between two persons is defined as follows:
$a R b left right double arrow a$ and $b$ born in different months. Then, $R$ is

If $A equals open curly brackets 1 comma blank 2 comma blank 3 comma blank 4 close curly brackets comma$ then the number of subsets of set $A$ containing element 3, is

If $A equals open curly brackets 1 comma blank 2 comma blank 3 comma blank 4 close curly brackets comma$ then the number of subsets of set $A$ containing element 3, is

If $A equals left curly bracket 4 comma blank 6 comma blank 10 comma blank 12 right curly bracket$ and $R$ is a relation defined on $A$ as “two elements are related iff they have exactly one common factor other than 1”. Then the relation $R$ is

If $A equals left curly bracket 4 comma blank 6 comma blank 10 comma blank 12 right curly bracket$ and $R$ is a relation defined on $A$ as “two elements are related iff they have exactly one common factor other than 1”. Then the relation $R$ is

parallel

$X$ is the set of all residents in a colony and $R$ is a relation defined on $X$ as follows:
“Two persons are related iff they speak the same language”
The relation $R$ is

$X$ is the set of all residents in a colony and $R$ is a relation defined on $X$ as follows:
“Two persons are related iff they speak the same language”
The relation $R$ is

If $R subset of A cross times B$ and $S subset of B cross times C$ be relations, then $open parentheses S o R close parentheses to the power of negative 1 end exponent equals$

If $R subset of A cross times B$ and $S subset of B cross times C$ be relations, then $open parentheses S o R close parentheses to the power of negative 1 end exponent equals$

If $A$ and $B$ are two sets such that $n open parentheses A intersection stack B with minus on top close parentheses equals 9 comma n open parentheses stack A with minus on top intersection B close parentheses equals 10$ and $n open parentheses A union B close parentheses equals 24 comma$ then $n open parentheses A cross times B close parentheses equals$

If $A$ and $B$ are two sets such that $n open parentheses A intersection stack B with minus on top close parentheses equals 9 comma n open parentheses stack A with minus on top intersection B close parentheses equals 10$ and $n open parentheses A union B close parentheses equals 24 comma$ then $n open parentheses A cross times B close parentheses equals$

parallel

If $R equals left curly bracket open parentheses a comma b close parentheses colon a plus b equals 4 right curly bracket$ is a relation on $N comma$ then $R$ is

If $R equals left curly bracket open parentheses a comma b close parentheses colon a plus b equals 4 right curly bracket$ is a relation on $N comma$ then $R$ is

If $P$ is the set of all parallelograms, and $T$ is the set of all trapeziums, then $P intersection T$ is

If $P$ is the set of all parallelograms, and $T$ is the set of all trapeziums, then $P intersection T$ is

If a set has 13 elements and $R$ is a reflexive relation on $A$ with $n$ elements, then

If a set has 13 elements and $R$ is a reflexive relation on $A$ with $n$ elements, then

parallel

Let $A$ be the non-void set of the children in a family. The relation $´ x$ is a brother of $y ´$ on $A$ is

Let $A$ be the non-void set of the children in a family. The relation $´ x$ is a brother of $y ´$ on $A$ is

If $A equals open curly brackets a comma b comma blank c close curly brackets comma blank B equals left curly bracket b comma blank c comma blank d right curly bracket$ and $C equals left curly bracket a comma blank d comma blank c right curly bracket$ , then $open parentheses A minus B close parentheses cross times left parenthesis B intersection C right parenthesis$ is equal to

If $A equals open curly brackets a comma b comma blank c close curly brackets comma blank B equals left curly bracket b comma blank c comma blank d right curly bracket$ and $C equals left curly bracket a comma blank d comma blank c right curly bracket$ , then $open parentheses A minus B close parentheses cross times left parenthesis B intersection C right parenthesis$ is equal to

Set builder form of the relation
$R equals open curly brackets open parentheses negative 2 comma negative 7 close parentheses comma open parentheses negative 1 comma negative 4 close parentheses comma open parentheses 0 comma negative 1 close parentheses comma open parentheses 1 comma blank 2 close parentheses comma open parentheses 2 comma blank 5 close parentheses close curly brackets$ is

Set builder form of the relation
$R equals open curly brackets open parentheses negative 2 comma negative 7 close parentheses comma open parentheses negative 1 comma negative 4 close parentheses comma open parentheses 0 comma negative 1 close parentheses comma open parentheses 1 comma blank 2 close parentheses comma open parentheses 2 comma blank 5 close parentheses close curly brackets$ is

parallel

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