Advertisements
Advertisements
प्रश्न
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
उत्तर
Given expression = (1 + x )2n
Coefficient of second term = 2nC1
Coefficient of third term = 2nC2
And coefficient of fourth term = 2nC3
As the given condition
2nC1. 2nC2 and 2nC3 are in A.P.
∴ 2nC2 – 2nC1 = 2nC3 – 2nC2
⇒ `2 * ""^(2n)"C"_2 = ""^(2n)"C"_1 + ""^(2n)"C"_3`
⇒ `2 * (2n!)/(2!(2n - 2)!) = (2n!)/((2n - 1)!) + (2n!)/(3!(2n - 3)!)`
⇒ `2[(2n(2n - 1)(2n - 2)!)/(2 xx 1 xx (2n - 2)!)] = (2n(2n - 1)!)/((2n - 1)!) + (2n(2n - 1)(2n - 2)(2n - 3)!)/(3 xx 2 xx 1 xx (2n - 3)!)`
⇒ n(2n – 1) = `n + (n(2n - 1)(2n - 2))/6`
⇒ 2n – 1 = `1 + ((2n - 1)(2n - 2))/6`
⇒ 12n – 6 = 6 + 4n2 – 4n – 2n + 2
⇒ 12n – 12 = 4n2 – 6n + 2
⇒ 4n2 – 6n – 12n + 2 + 12 = 0
⇒ 4n2 – 18n + 14 = 0
⇒ 2n2 – 9n + 7 = 0
Hence proved.
APPEARS IN
संबंधित प्रश्न
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively.
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Find an approximation of (0.99)5 using the first three terms of its expansion.
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Which of the following is larger? 9950 + 10050 or 10150
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O2 – E2 = (x2 – a2)n
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
