Advertisements
Advertisements
प्रश्न
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find n.
Advertisements
उत्तर
\[\text{ Suppose r, } \left( r + 1 \right) \text{ and } \left( r + 2 \right) \text{ are three consecutive terms in the given expansion } . \]
\[\text { The coefficients of these terms are } ^{n}{}{C}_{r - 1} , ^{n}{}{C}_r \text{ and } ^ {n}{}{C}_{r + 1} . \]
\[\text{ According to the question, } \]
\[ ^{n}{}{C}_{r - 1} = 76\]
\[^{n}{}{C}_r = 95\]
\[ ^{n}{}{C}_{r + 1} = 76\]
\[ \Rightarrow ^{n}{}{C}_{r - 1} = ^{n}{}{C}_{r + 1} \]
\[ \Rightarrow r - 1 + r + 1 = n [\text{ If } ^{n}{}{C}_r = ^{n}{}{C}_s \Rightarrow r = s \text{ or } r + s = n]\]
\[ \Rightarrow r = \frac{n}{2}\]
\[ \therefore \frac{^{n}{}{C}_r}{^{n}{}{C}_{r - 1}} = \frac{95}{76}\]
\[ \Rightarrow \frac{n - r + 1}{r} = \frac{95}{76}\]
\[ \Rightarrow \frac{\frac{n}{2} + 1}{\frac{n}{2}} = \frac{95}{76}\]
\[ \Rightarrow 38n + 76 = \frac{95n}{2} \]
\[ \Rightarrow \frac{19n}{2} = 76\]
\[ \Rightarrow n = 8\]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – y)6
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n–1 .
Find the middle terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms in the expansion of:
(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Find the middle terms(s) in the expansion of:
(i) \[\left( x - \frac{1}{x} \right)^{10}\]
Find the middle terms(s) in the expansion of:
(iii) \[\left( 1 + 3x + 3 x^2 + x^3 \right)^{2n}\]
Find the middle terms(s) in the expansion of:
(x) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
Find the term independent of x in the expansion of the expression:
(v) \[\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, find x, a, n.
Write the middle term in the expansion of \[\left( x + \frac{1}{x} \right)^{10}\]
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
In the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9\] , the term independent of x is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find the middle term in the expansion of `(2ax - b/x^2)^12`.
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
Find the term independent of x in the expansion of `(3x - 2/x^2)^15`
Find the middle term (terms) in the expansion of `(x/a - a/x)^10`
Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal.
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
The number of terms in the expansion of [(2x + y3)4]7 is 8.
If n is the number of irrational terms in the expansion of `(3^(1/4) + 5^(1/8))^60`, then (n – 1) is divisible by ______.
If the coefficient of x10 in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5kl, where l, k ∈ N and l is coprime to 5, then k is equal to ______.
The term independent of x in the expansion of `[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`, x ≠ 1 is equal to ______.
