Home
Class 12
MATHS
If the 3rd, 4th , 5th and 6th term in th...

If the 3rd, 4th , 5th and 6th term in the expansion of `(x+alpha)^n` be, respectively, `a ,b ,ca n dd ,` prove that `(b^2-a c)/(c^2-b d)=(5a)/(3c)dot`

Text Solution

Verified by Experts

We know that
`(T_(r+1))/(T_(r)), (n-r+1)/(r)(alpha)/(x)`
`T_(3)=a,T_(4)=b,T_(5)=c,T_(6)=d`
Putting `r = 3, 4,5` in the abvoe we get
`(T_(4))/(T_(3)) = (n-2)/(3)(alpha)/(x) = b/a`
`(T_(5))/(T_(4)) = (n-3)/(4)(alpha)/(x) = c/b`
`(T_(6))/(T_(5)) = (n-4)/(5) (alpha)/(x) = d/c`
We have to prove
`(b^(2)-ac)/(c^(2)-bd) = (5a)/(3c)`
or `((b)/(c)-(a)/(b))/((c)/(b)-(d)/(c)) = (5a)/(3c)`
Now, `((b)/(c) - (a)/(b))/((c)/(b)-(d)/(c)) = ((4x)/((n-3)alpha)-(3x)/((n-2)alpha))/(((n-3)alpha)/(4x) - ((n-4)alpha)/(5x))`
`= (x^(2))/(alpha^(2))(4xx5)/((n-2)(n-3))(4n-8-3n+9)/(5n-15-4n+16)`
`= (x^(2))/(alpha^(2))(4xx5)/((n-2)(n-3))`
Also, `(5a)/(3c) = (x^(2))/(alpha^(2))(4xx5)/((n-2)(n-3))`
`:. L.H.S. = R.H.S.`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.6|10 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.7|9 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.4|13 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|20 Videos

  • BINOMIAL THEORM

    CENGAGE|Exercise Question Bank|31 Videos

Similar Questions

Explore conceptually related problems

If 2nd, 3rd and 4th tems in the expansion of (x+y)^n be 240,720 and 1080 respectively, find x,y and n.

If the 2 nd,3 rd and 4 th terms in the expansion of (x+a)^(n) are 240,720 and 1080 respectively,find x,a,n .

If the 5 th term is 24 xx the 3rd term in the expansion of (1+x)^(n), then x=

If n be a positive integer and if the 3rd, 4th, 5th and 6th terms in the expansion of (x+A)^n, when expanded in ascending powers of x, be a, b, c and d respectively show that, (b^2-ac)/(c^2-bd)=(5a)/(3c)

If 6th , 7th , 8th, and 9th terms of (x+ y)^n are a,b,c and d respectively , then prove that : (b^2 - ac)/(c^2 -bd) =4/3. a/c

If 3rd,4th, 5th terms in the expansion of (a+x)^(n) be 84,280 and 560, Find x, a and n .

Find n if the coefficients of 4th the 13th terms in the expansion of (a+b)^n are equal.

If the 3^(rd) , 4^(th) and 5^(th) terms in the expansion of (x + a)^(n) are 84, 280 and 560 respectively then find the values of a, x and n.

If the 6th,7th, 8th terms in the expansion of (x+y)^(n) be 112,7 and 1/4 find x,y and n.

If in any binomial expansion a,b,c and d be the 6th,7th,8th and 9th terms respectively, prove that (b^(2)-ac)/(c^(2)-bd)=(4a)/(3c)