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
Find the coefficient of a5b7 in (a – 2b)12
Find the 4th term in the expansion of (x – 2y)12 .
Find the 13th term in the expansion of `(9x - 1/(3sqrtx))^18 , x != 0`
Find the middle terms in the expansions of `(x/3 + 9y)^10`
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 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:
(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 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:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Find the term independent of x in the expansion of the expression:
(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
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)2n are in A.P., show that \[2 n^2 - 9n + 7 = 0\]
If in the expansion of (1 + x)n, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where \[p \neq q\]
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 a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
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
In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is 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 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 in the expansion of `(3x - 2/x^2)^15`
Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
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`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term 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 ______.
