Advertisements
Advertisements
प्रश्न
The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n–1 are in the ratio ______.
पर्याय
1 : 2
1 : 3
3 : 1
2 : 1
उत्तर
The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n–1 are in the ratio 2 : 1
Explanation:
General Term `"T"_(r + 1) = ""^"n""C"_r x^(n - r) y^r`
In the expansion of (1 + x)2n
We get `"T"_(r + 1) = ""^(2n)"C"_r x^r`
To get the coefficient of xn
Put r = n
∴ Coefficient of xn = 2nCn
In the expansion of (1 + x)2n–1
We get `"T"_(r + 1) = ""^(2n - 1)"C"_r x^r`
∴ Coefficient of xn = `""^(2n - 1)"C"_(n - 1)`
The required ratio is `(""^(2n)"C"_n)/(""^(2n - 1)"C"_(n - 1))`
= `((2n!)/(n!(n!)))/(((2n - 1)!)/((n - 1)!(2n - 1 - n + 1)!))`
= `((22n!)/(n!n!))/(((2n - 1)!)/((n - 1)(n!))`
= `(2n!)/(n!n!) xx ((n - 1)!*n!)/((2n - 1)!)`
= `(2n(2n - 1)!)/(n!n(n - 1)!) xx ((n - 1)!*n!)/((2n - 1)!)`
= `2/1`
= 2 : 1
APPEARS IN
संबंधित प्रश्न
Using binomial theorem, indicate which is larger (1.1)10000 or 1000.
Find the ratio of the coefficients of xp and xq in the expansion of \[\left( 1 + x \right)^{p + q}\] .
Write last two digits of the number 3400.
If a and b are the coefficients of xn in the expansion of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, find \[\frac{a}{b}\]
If \[\left( 1 - x + x^2 \right)^n = a_0 + a_1 x + a_2 x^2 + . . . + a_{2n} x^{2n}\] , find the value of \[a_0 + a_2 + a_4 + . . . + a_{2n}\] .
If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then ris equal to
The coefficient of \[x^{- 17}\] in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] is
If the coefficients of 2nd, 3rd and the 4th terms in the expansion of (1 + x)n are in A.P., then value of n is ______.
If A and B are coefficient of x n in the expansions of (1 + x)2n and (1 + x)2n–1 respectively, then `A/B` equals ______.
The largest coefficient in the expansion of (1 + x)30 is ______.
The ratio of the coefficients of xp and xq in the expansion of (1 + x)p + q is ______.
If 2515 is divided by 13, the reminder is ______.
The sum of the series `sum_(r = 0)^10 ""^20C_r` is `2^19 + (""^20C_10)/2`
The expression 79 + 97 is divisible by 64.
