Advertisements
Advertisements
प्रश्न
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
विकल्प
3rd and 4th
4th and 5th
5th and 6th
6th and 7th
Advertisements
उत्तर
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are 5th and 6th.
Explanation:
Let rth and (r + 1)th be two successive terms in the expansion (1 + x)24
∴ `"T"_(r + 1) = ""^24"C"_r * x^r`
`"T"_(r + 2) = "T"_(r + 1 + 1) = ""^24"C"_(r + 1) x^(r + 1)`
We have `(""^24"C"_r)/(""^24"C"_(r + 1)) = 1/4`
⇒ `((24!)/(r!(24 - r)!))/((24!)/((r + 1)!(24 - r - 1)!)) = 1/4`
⇒ `(24!)/(r!(24 - r)!) xx ((r - 1)!(24 - r - 1)!)/(24!) = 1/4`
⇒ `((r + 1) * r!(24 - r - 1)!)/(r!(24 - r)(24 - r - 1)!) = 1/4`
⇒ `(r + 1)/(24 - r) = 1/4`
⇒ 4r + 4 = 24 – r
⇒ 5r = 20
⇒ r = 4
∴ T4+1 = T5 and T4+2 = T6
APPEARS IN
संबंधित प्रश्न
Expand the expression: (1– 2x)5
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Expand the expression: (2x – 3)6
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate f the following:
(101)4
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
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?
Expand the following (1 – x + x2)4
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
If a1, a2, a3 and a4 are the coefficient of any four consecutive terms in the expansion of (1 + x)n, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Find the coefficient of x15 in the expansion of (x – x2)10.
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
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 4OE = (x + a)2n – (x – a)2n
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
The number of terms in the expansion of (x + y + z)n ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
