Advertisements
Advertisements
Question
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find n.
Advertisements
Solution
\[\text{ Suppose r, } \left( r + 1 \right) \text{ and } \left( r + 2 \right) \text{ are three consecutive terms in the given expansion } . \]
\[\text { The coefficients of these terms are } ^{n}{}{C}_{r - 1} , ^{n}{}{C}_r \text{ and } ^ {n}{}{C}_{r + 1} . \]
\[\text{ According to the question, } \]
\[ ^{n}{}{C}_{r - 1} = 76\]
\[^{n}{}{C}_r = 95\]
\[ ^{n}{}{C}_{r + 1} = 76\]
\[ \Rightarrow ^{n}{}{C}_{r - 1} = ^{n}{}{C}_{r + 1} \]
\[ \Rightarrow r - 1 + r + 1 = n [\text{ If } ^{n}{}{C}_r = ^{n}{}{C}_s \Rightarrow r = s \text{ or } r + s = n]\]
\[ \Rightarrow r = \frac{n}{2}\]
\[ \therefore \frac{^{n}{}{C}_r}{^{n}{}{C}_{r - 1}} = \frac{95}{76}\]
\[ \Rightarrow \frac{n - r + 1}{r} = \frac{95}{76}\]
\[ \Rightarrow \frac{\frac{n}{2} + 1}{\frac{n}{2}} = \frac{95}{76}\]
\[ \Rightarrow 38n + 76 = \frac{95n}{2} \]
\[ \Rightarrow \frac{19n}{2} = 76\]
\[ \Rightarrow n = 8\]
APPEARS IN
RELATED QUESTIONS
Find the coefficient of a5b7 in (a – 2b)12
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Find the middle terms in the expansions of `(x/3 + 9y)^10`
Find the middle term in the expansion of:
(i) \[\left( \frac{2}{3}x - \frac{3}{2x} \right)^{20}\]
Find the middle term in the expansion of:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
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:
(i) \[\left( x - \frac{1}{x} \right)^{10}\]
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
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 p is a real number and if the middle term in the expansion of \[\left( \frac{p}{2} + 2 \right)^8\] is 1120, find p.
Find the sum of the coefficients of two middle terms in the binomial expansion of \[\left( 1 + x \right)^{2n - 1}\]
If in the expansion of (a + b)n and (a + b)n + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is
The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\] is
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
If in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] , \[x^{- 17}\] occurs in rth term, then
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 term independent of x in the expansion of `(3x - 2/x^2)^15`
If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
Middle term in the expansion of (a3 + ba)28 is ______.
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 is ______.
The sum of the co-efficients of all even degree terms in x in the expansion of `(x + sqrt(x^3 - 1))^6 + (x - sqrt(x^3 - 1))^6, (x > 1)` is equal to ______.
The coefficient of y49 in (y – 1)(y – 3)(y – 5) ...... (y – 99) 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 ______.
The sum of the real values of x for which the middle term in the binomial expansion of `(x^3/3 + 3/x)^8` equals 5670 is ______.
