Advertisements
Advertisements
प्रश्न
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, find x, a, n.
Advertisements
उत्तर
\[\text{ The 6th, 7th and 8th terms in the expansion of } (x + a )^n \text{ are } ^{n}{}{C}_5 x^{n - 5} a^5 , ^{n}{}{C}_6 x^{n - 6} a^6 \text{ and } ^{n}{}{C}_7 x^{n - 7} a^7 .\]
According to the question,
\[^{n}{}{C}_5 x^{n - 5} a^5 = 112\]
\[ ^{n}{}{C}_6 x^{n - 6} a^6 = 7\]
\[ ^{n}{}{C}_7 x^{n - 7} a^7 = \frac{1}{4}\]
\[\text{ Now } , \]
\[\frac{^{n}{}{C}_6 x^{n - 6} a^6}{^{n}{}{C}_5 x^{n - 5} a^5} = \frac{7}{112}\]
\[ \Rightarrow \frac{n - 6 + 1}{6} x^{- 1} a = \frac{1}{16}\]
\[ \Rightarrow \frac{a}{x} = \frac{3}{8n - 40} . . . \left( 1 \right)\]
\[\text{ Also, } \]
\[\frac{^{n}{}{C}_7 x^{n - 7} a^7}{^{n}{}{C}_6 x^{n - 6} a^6} = \frac{1/4}{7}\]
\[ \Rightarrow \frac{n - 7 + 1}{7} x^{- 1} a = \frac{1}{28}\]
\[ \Rightarrow \frac{a}{x} = \frac{1}{4n - 24} . . . \left( 2 \right)\]
\[\text{ From } \left( 1 \right) \text{ and } \left( 2 \right), \text{ we get: } \]
\[\frac{3}{8n - 40} = \frac{1}{4n - 24}\]
\[ \Rightarrow \frac{3}{2n - 10} = \frac{1}{n - 6}\]
\[ \Rightarrow n = 8\]
\[\text{ Putting in eqn } \left( 1 \right) \text{ we get} \]
\[ \Rightarrow a = x\]
\[\text{ Now, } ^{8}{}{C}_5 x^{8 - 5} \left( \frac{x}{8} \right)^5 = 112\]
\[ \Rightarrow \frac{56 x^8}{8^5} = 112\]
\[ \Rightarrow x^8 = 4^8 \]
\[ \Rightarrow x = 4\]
\[\text{ By putting the value of x and n in } \left( 1 \right) \text{ we get} \]
\[a = \frac{1}{2}\]
\[a = 3 \text{ and } x = 2\]
APPEARS IN
संबंधित प्रश्न
Find the coefficient of a5b7 in (a – 2b)12
Find the 13th term in the expansion of `(9x - 1/(3sqrtx))^18 , x != 0`
Find the middle term in the expansion of:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
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:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms in the expansion of:
(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Find the middle terms in the expansion of:
(iv) \[\left( x^4 - \frac{1}{x^3} \right)^{11}\]
Find the term independent of x in the expansion of the expression:
(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
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) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
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 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 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 the term free from x in the expansion of \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\] is 405, find the value of k.
Write the middle term in the expansion of \[\left( x + \frac{1}{x} \right)^{10}\]
Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
Write the total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .
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 \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
In the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9\] , the term independent of x is
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
The ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is ______.
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
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`
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 ______.
The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` 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 middle term in the expansion of (1 – 3x + 3x2 – x3)6 is ______.
If the coefficient of x10 in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5kl, where l, k ∈ N and l is coprime to 5, then k is equal to ______.
