Assertion -Let f n -r-1nr4- then -r-1nr n-r 3 is equal to n n n-1 -2 2-f n Reason- -r-1nr n-1 3-r-1n n-1 r3

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Standard XII

Mathematics

AM,GM,HM Inequality

Assertion :Le...

Question

Assertion :Let $f (n) = n \sum r = 1 r^{4}, t h e n n \sum r = 1 r {(n - r)}^{3}$ is equal to $n {(\frac{n (n + 1)}{2})}^{2} - f (n)$ Reason: $n \sum r = 1 r {(n - 1)}^{3} = n \sum r = 1 (n - 1) r^{3}$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$n \sum r = 1 r (n - r)^{3} = n \sum r = 1 (n - r) r^{3}$
$= n \sum r = 1 (n r^{3} - r^{4})$
$= n n \sum r = 1 r^{3} - n \sum r = 1 r^{4} = n {\frac{n (n + 1)}{2}}^{2} - f (n)$
Ans: A

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Similar questions

This question consists of two statements,namely,Assertion (A) and Reason (R).For selecting the correct answer,use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

The correct answer is :(a) /(b) /(c) /(d)

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Reason (R) There is a repulsion between the two bulky (- R) groups.
(a) Assertion and reason both are correct and reason is correct explanation of assertion.
(b) Assertion and reason both are wrong statements.
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(d) Assertion is wrong statement but reason is correct statement.
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Assertion (A) Order of the reaction can be zero or fractional.

Reason (R) We cannot determine order from balanced chemical equation.

(a) Both assertion and reason are correct and the reason is correct explanation of assertion
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(c) Assertion is correct, but reason is incorrect
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This question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(d) Assertion (A) is false and Reason (R) is true.

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Assertion :Let f(n)=n∑r=1r4, then n∑r=1r(n−r)3 is equal to n(n(n+1)2)2−f(n) Reason: n∑r=1r(n−1)3=n∑r=1(n−1)r3

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertionn∑r=1r(n−r)3=n∑r=1(n−r)r3=n∑r=1(nr3−r4)=nn∑r=1r3−n∑r=1r4=n{n(n+1)2}2−f(n)Ans: A

Assertion :Let $f (n) = n \sum r = 1 r^{4}, t h e n n \sum r = 1 r {(n - r)}^{3}$ is equal to $n {(\frac{n (n + 1)}{2})}^{2} - f (n)$ Reason: $n \sum r = 1 r {(n - 1)}^{3} = n \sum r = 1 (n - 1) r^{3}$

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$n \sum r = 1 r (n - r)^{3} = n \sum r = 1 (n - r) r^{3}$
$= n \sum r = 1 (n r^{3} - r^{4})$
$= n n \sum r = 1 r^{3} - n \sum r = 1 r^{4} = n {\frac{n (n + 1)}{2}}^{2} - f (n)$
Ans: A