Advertisements
Advertisements
Question
If a and b are coefficients of xn in the expansions of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, then write the relation between a and b.
Advertisements
Solution
\[\text{ Coefficient of } x^n \text{ in the expansion} (1 + x )^{2n} =^{2n}{}{C}_n = a\]
\[\text{ Coefficient of } x^n \text{ in the expansion} (1 + x )^{2n - 1} = ^{2n - 1}{}{C}_n = b\]
\[\text{ Now, we have:} \]
\[ ^{2n}{}{C}_n = \frac{2n!}{n! . n!} = \frac{2n(2n - 1)!}{n\left( n - 1 \right)! n!} . . . \left( 1 \right)\]
\[ \text{ and } ^{2n - 1}{}{C}_n = \frac{(2n - 1)!}{n!(n - 1)!} . . . \left( 2 \right)\]
\[\text{ Dividing equation } \left( 1 \right) \text{ by } \left( 2 \right), \text{ we get } \]
\[ \Rightarrow \frac{^{2n}{}{C}_n}{^{2n - 1}{}{C}_n} = \frac{2n(2n - 1)! n! (n - 1)!}{n\left( n - 1 \right)! n! (2n - 1)!}\]
\[ \Rightarrow \frac{a}{b} = 2\]
\[ \Rightarrow a = 2b\]
APPEARS IN
RELATED QUESTIONS
Using binomial theorem, write down the expansions .
(i) \[\left( 2x + 3y \right)^5\]
Using binomial theorem, write down the expansions :
(ii) \[\left( 2x - 3y \right)^4\]
Using binomial theorem, write down the expansions .
(iii) \[\left( x - \frac{1}{x} \right)^6\]
Using binomial theorem, write down the expansions :
(iv) \[\left( 1 - 3x \right)^7\]
Using binomial theorem, write down the expansions :
(vii) \[\left( \sqrt[3]{x} - \sqrt[3]{a} \right)^6\]
Using binomial theorem, write down the expansions :
(x) \[\left( 1 - 2x + 3 x^2 \right)^3\]
Evaluate the
(i)\[\left( \sqrt{x + 1} + \sqrt{x - 1} \right)^6 + \left( \sqrt{x + 1} - \sqrt{x - 1} \right)^6\]
Evaluate the
(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]
Evaluate the
(iii)\[\left( 1 + 2 \sqrt{x} \right)^5 + \left( 1 - 2 \sqrt{x} \right)^5\]
Evaluate the
(v) \[\left( 3 + \sqrt{2} \right)^5 - \left( 3 - \sqrt{2} \right)^5\]
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
Evaluate the
(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4\]
Find \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\] . Hence, or otherwise evaluate \[\left( \sqrt{2} + 1 \right)^6 + \sqrt{2} - 1^6\] .
Using binomial theorem evaluate :
(i) (96)3
Using binomial theorem evaluate .
(iii) (101)4
Using binomial theorem evaluate .
(iv) (98)5
Using binomial theorem, prove that \[3^{2n + 2} - 8n - 9\] is divisible by 64, \[n \in N\] .
Find the coefficient of:
(i) x10 in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{20}\]
Find the coefficient of:
(ii) x7 in the expansion of \[\left( x - \frac{1}{x^2} \right)^{40}\]
Find the coefficient of:
(v) \[x^m\] in the expansion of \[\left( x + \frac{1}{x} \right)^n\]
Find the coefficient of:
(vi) x in the expansion of \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]
Find the coefficient of:
(vii) \[a^5 b^7\] in the expansion of \[\left( a - 2b \right)^{12}\]
Which term in the expansion of \[\left\{ \left( \frac{x}{\sqrt{y}} \right)^{1/3} + \left( \frac{y}{x^{1/3}} \right)^{1/2} \right\}^{21}\] contains x and y to one and the same power?
Does the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)\] contain any term involving x9?
If a and b denote respectively the coefficients of xm and xn in the expansion of \[\left( 1 + x \right)^{m + n}\], then write the relation between a and b.
The term without x in the expansion of \[\left( 2x - \frac{1}{2 x^2} \right)^{12}\] is
The coefficient of x4 in \[\left( \frac{x}{2} - \frac{3}{x^2} \right)^{10}\] is
The coefficient of \[\frac{1}{x}\] in the expansion of \[\left( 1 + x \right)^n \left( 1 + \frac{1}{x} \right)^n\] is
If the coefficients of x2 and x3 in the expansion of (3 + ax)9 are the same, then the value of a is
