If an the expansion of \[\left( 1 + x \right)^{15}\] , the coefficients of \[\left( 2r + 3 \right)^{th}\text{ and } \left( r - 1 \right)^{th}\] terms are equal, then the value of r is

5 \[\text{ Coefficients of (2r + 3)th and (r - 1)th terms in the given expansion are } ^{15}{}{C}_{2r + 2} \text{ and }^{15}{}{C}_{r - 2 .} \] \[\text{ Thus, we have } \] \[ ^{15}{}{C}_{2r + 2} = ^{15}{}{C}_{r - 2} \] \[ \Rightarrow 2r + 2 = r - 2 \text{ or } 2r + 2 + r - 2 = 15 \left[ \because \text{ if } {}^n C_x =^n C_y \Rightarrow x = y \text{ or } x + y = n \right] \] \[ \Rightarrow r = - 4 \text{ or } r = 5\] \[\text{ Neglecting the negative value, We have } \] \[r = 5\]

If an the Expansion of ( 1 + X ) 15 , the Coefficients of ( 2 R + 3 ) T H and ( R − 1 ) T H Terms Are Equal, Then the Value of R is (A) 5 (B) 6 (C) 4 (D) 3 - Mathematics

प्रश्न

If an the expansion of ${(1 + x)}^{15}$ , the coefficients of ${(2 r + 3)}^{t h} and {(r - 1)}^{t h}$ terms are equal, then the value of r is

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उत्तर

${Coefficients of (2r + 3)th and (r - 1)th terms in the given expansion are}^{15} C_{2 r + 2} {and}^{15} C_{r - 2 .}$

$Thus, we have$

$^{15} C_{2 r + 2} =^{15} C_{r - 2}$

$\Rightarrow 2 r + 2 = r - 2 or 2 r + 2 + r - 2 = 15 [∵ if^{n} C_{x} =^{n} C_{y} \Rightarrow x = y or x + y = n]$

$\Rightarrow r = - 4 or r = 5$

$Neglecting the negative value, We have$

$r = 5$

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General and Middle Terms

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Q 9 Q 8 Q 10

अध्याय 18: Binomial Theorem - Exercise 18.4 [पृष्ठ ४६]

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आरडी शर्मा Mathematics [English] Class 11

अध्याय 18 Binomial Theorem
Exercise 18.4 | Q 9 | पृष्ठ ४६

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General and Middle Terms

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If an the Expansion of ( 1 + X ) 15 , the Coefficients of ( 2 R + 3 ) T H and ( R − 1 ) T H Terms Are Equal, Then the Value of R is (A) 5 (B) 6 (C) 4 (D) 3 - Mathematics

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