Advertisements
Advertisements
प्रश्न
If the 2nd, 3rd and 4th terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, find x, a, n.
Advertisements
उत्तर
\[\text{ In the expansion of } \left( x + a \right)^n , \text{ the 2nd, 3rd and 4th terms are } ^{n}{}{C}_1 x^{n - 1} a^1 , ^{n}{}{C}_2 x^{n - 2} a^2 \text{ and } ^{n}{}{C}_3 x^{n - 3} a^3 , \ \text{ respectively } . \]
\[\text{ According to the question } , \]
\[ ^{n}{}{C}_1 x^{n - 1} a^1 = 240 \]
\[ ^{n}{}{C}_2 x^{n - 2} a^2 = 720\]
\[^{n}{}{C}_3 x^{n - 3} a^3 = 1080\]
\[ \Rightarrow \frac{^{n}{}{C}_2 x^{n - 2} a^2}{^{n}{}{C}_1 x^{n - 1} a^1} = \frac{720}{240}\]
\[ \Rightarrow \frac{n - 1}{2x}a = 3\]
\[ \Rightarrow \frac{a}{x} = \frac{6}{n - 1} . . . \left( 1 \right)\]
\[\text{ Also } , \]
\[\frac{^{n}{}{C}_3 x^{n - 3} a^3}{^{n}{}{C}_2 x^{n - 2} a^2} = \frac{1080}{720}\]
\[ \Rightarrow \frac{n - 2}{3x}a = \frac{3}{2}\]
\[ \Rightarrow \frac{a}{x} = \frac{9}{2n - 4} . . . \left( 2 \right)\]
\[\text{ Using } \left( 1 \right) \text{ and } \left( 2 \right) \text{ we get } \]
\[\frac{6}{n - 1} = \frac{9}{2n - 4}\]
\[ \Rightarrow n = 5\]
\[\text{ Putting in eqn } \left( 1 \right) \text{ we get } \]
\[ \Rightarrow 2a = 3x\]
\[\text{ Now } , ^{5}{}{C}_1 x^{5 - 1} \left( \frac{3}{2}x \right) = 240\]
\[ \Rightarrow 15 x^5 = 480\]
\[ \Rightarrow x^5 = 32\]
\[ \Rightarrow x = 2\]
\[\text{ By putting the value of x and n in} \left( 1 \right) \text{ we get} \]
\[a = 3\]
APPEARS IN
संबंधित प्रश्न
Find the coefficient of x5 in (x + 3)8
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:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
Find the middle terms in the expansion of:
(iv) \[\left( x^4 - \frac{1}{x^3} \right)^{11}\]
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:
(ii) \[\left( 1 - 2x + x^2 \right)^n\]
Find the middle terms(s) in the expansion of:
(vi) \[\left( \frac{x}{3} + 9y \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(ii) \[\left( 2x + \frac{1}{3 x^2} \right)^9\]
Find the term independent of x in the expansion of the expression:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Find the term independent of x in the expansion of the expression:
(iv) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
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 in the expansion of (1 + x)n, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where \[p \neq q\]
If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
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
In the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9\] , the term independent of x is
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[\left( x + a \right)^n\] are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] 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
Find the middle term in the expansion of `(2ax - b/x^2)^12`.
Find the middle term (terms) in the expansion of `(x/a - a/x)^10`
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 n in the binomial `(root(3)(2) + 1/(root(3)(3)))^n` if the ratio of 7th term from the beginning to the 7th term from the end is `1/6`
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is ______.
The number of terms in the expansion of [(2x + y3)4]7 is 8.
The last two digits of the numbers 3400 are 01.
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 ______.
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 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 ______.
