Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[\text{ We have: } \]
\[ T_1 = 729, T_2 = 7290 \text{ and } T_3 = 30375\]
\[\text{ Now,} \]
\[ ^{n}{}{C}_0 a^n b^0 = 729\]
\[ \Rightarrow a^n = 729\]
\[ \Rightarrow a^n = 3^6 \]
\[ ^{n}{}{C}_1 a^{n - 1} b^1 = 7290\]
\[^{n}{}{C}_2 a^{n - 2} b^2 = 30375\]
\[\text{ Also, } \]
\[\frac{^{n}{}{C}_2 a^{n - 2} b^2}{^{n}{}{C}_1 a^{n - 1} b^1} = \frac{30375}{7290}\]
\[ \Rightarrow \frac{n - 1}{2} \times \frac{b}{a} = \frac{25}{6} . . . (i)\]
\[ \Rightarrow \frac{(n - 1)b}{a} = \frac{25}{3}\]
\[\text{ And } , \]
\[\frac{^{n}{}{C}_1 a^{n - 1} b^1}{^{n}{}{C}_0 a^n b^0} = \frac{7290}{729}\]
\[ \Rightarrow \frac{nb}{a} = \frac{10}{1} . . . (ii)\]
\[\text{ On dividing (ii) by (i), we get } \]
\[\frac{\frac{nb}{a}}{\frac{(n - 1)b}{a}} = \frac{10 \times 3}{25}\]
\[ \Rightarrow \frac{n}{n - 1} = \frac{6}{5}\]
\[ \Rightarrow n = 6\]
\[\text{ Since } , a^6 = 3^6 \]
\[\text{ Hence,} a = 3\]
\[\text{ Now } , \frac{nb}{a} = 10\]
\[ \Rightarrow b = 5\]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Find the 4th term in the expansion of (x – 2y)12 .
In the expansion of (1 + a)m + n, prove that coefficients of am and an are equal.
The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1)n are in the ratio 1:3:5. Find n and r.
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of `(root4 2 + 1/ root4 3)^n " is " sqrt6 : 1`
Find the middle term in the expansion of:
(iii) \[\left( x^2 - \frac{2}{x} \right)^{10}\]
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 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:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
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:
(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:
(vii) \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]
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.
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.
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.
Write the middle term in the expansion of \[\left( x + \frac{1}{x} \right)^{10}\]
Write the total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .
The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\] is
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
The total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] after simplification is
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
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 middle term (terms) in the expansion of `(3x - x^3/6)^9`
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
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 ______.
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 ______.
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 ______.
