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
Find the coefficient of x5 in (x + 3)8
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Find the middle term in the expansion of:
(i) \[\left( \frac{2}{3}x - \frac{3}{2x} \right)^{20}\]
Find the middle terms in the expansion of:
(i) \[\left( 3x - \frac{x^3}{6} \right)^9\]
Find the middle terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms(s) in the expansion of:
(iv) \[\left( 2x - \frac{x^2}{4} \right)^9\]
Find the middle terms(s) in the expansion of:
(v) \[\left( x - \frac{1}{x} \right)^{2n + 1}\]
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find the middle terms(s) in the expansion of:
(ix) \[\left( \frac{p}{x} + \frac{x}{p} \right)^9\]
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:
(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:
(x) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6\]
If the coefficients of \[\left( 2r + 4 \right)\text{ th and } \left( r - 2 \right)\] th terms in the expansion of \[\left( 1 + x \right)^{18}\] are equal, find r.
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
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.
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
If in the expansion of (1 + x)n, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.
If 3rd, 4th 5th and 6th terms in the expansion of (x + a)n be respectively a, b, c and d, prove that `(b^2 - ac)/(c^2 - bd) = (5a)/(3c)`.
If the 2nd, 3rd and 4th terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, find x, a, n.
Find a, b and n in the expansion of (a + b)n, if the first three terms in the expansion are 729, 7290 and 30375 respectively.
If A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
If in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] , \[x^{- 17}\] occurs in rth term, then
The total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] after simplification is
If rth term is the middle term in the expansion of \[\left( x^2 - \frac{1}{2x} \right)^{20}\] then \[\left( r + 3 \right)^{th}\] term is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
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`
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.
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 4th term in the expansion of `(ax + 1/x)^n` is `5/2` then the values of a and n respectively are ______.
The middle term in the expansion of (1 – 3x + 3x2 – x3)6 is ______.
Let the coefficients of the middle terms in the expansion of `(1/sqrt(6) + βx)^4, (1 - 3βx)^2` and `(1 - β/2x)^6, β > 0`, common difference of this A.P., then `50 - (2d)/β^2` 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 ______.
