Advertisements
Advertisements
प्रश्न
If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
Advertisements
उत्तर
Let the binomial expansion be (x + y)n
Given,
`T_6 = ^nC_5 x^(n - 5) y^5 = a`
`T_7 = ^nC_6 x^(n - 6) y^6 = b`
`T_8 = ^nC_7 x^(n - 7) y^7 = c`
`T_9 = ^nC_8 x^(n - 8) y^8 = d`
To prove: `(b^2 - ac)/(c^2 - bd)= (4a)/(3c)`
⇒ `(b^2 - ac)/(a) = 4/3 [(c^2 - bd)/(c)]`
⇒ `1/b [(b^2 - ac)/a] = 4/3 [(c^2 - bd)/(bc)]`
⇒ `b/a - c/b = 4/3 [c/d - d/c]` ...(i)
Now, substituting the values of a, b, c and d, we get
`(""^nC_6 x^(n - 6) y^6)/(""^nC_5 x^(n - 5) y^5) - (""^nC_7 x^(n - 7) y^7)/(""^nC_6 x^(n - 6) y^6) = 4/3 [(""^nC_7 x^(n - 7) y^7)/(""^nC_6 x^(n - 6) y^6) - (""^nC_8 x^(n -8) y^8)/(""^nC_7 x ^(n - 7) y^7)]`
`[(""^nC_6)/(""^nC_5) - (""^nC_4)/(""^nC_6)]y/x = 4/3 y/x [(""^nC_7)/(""^nC_6) - (""^nC_8)/(""^nC_7)]`
We know that, `(""^nC_r)/(""^nC_(r - 1)) = (n - r + 1)/r`
∴ `[(n - 5)/6 - (n - 6)/7] = 4/3[(n - 6)/7 - (n - 7)/8]`
⇒ `(7n - 35 - 6n + 36)/42 = (8n - 48 - 7n + 49)/(3 xx 7 xx 2)`
⇒ `(n + 1)/42 = (n + 1)/42`
⇒ LHS = RHS
APPEARS IN
संबंधित प्रश्न
Find the coefficient of x5 in (x + 3)8
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 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:
(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:
(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:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
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 the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find 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.
Write the middle term in the expansion of `((2x^2)/3 + 3/(2x)^2)^10`.
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 \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .
If A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
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 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 ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is ______.
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`
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`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
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 sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
The last two digits of the numbers 3400 are 01.
The number of rational terms in the binomial expansion of `(4^(1/4) + 5^(1/6))^120` is ______.
