Advertisements
Advertisements
Question
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 \[a_0 + a_2 + a_4 + . . . + a_{2n}\] .
Solution
Putting x = 1 and −1 in \[\left( 1 - x + x^2 \right)^n = a_0 + a_1 x + a_2 x^2 + . . . + a_{2n} x^{2n}\] we get,
\[1 = a_0 + a_1 + a_2 + . . . + a_{2n} . . . (1)\]
and
\[3^n = a_0 - a_1 + a_2 - . . . + a_{2n} . . . (2)\]
Adding (1) and (2), we get
APPEARS IN
RELATED QUESTIONS
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 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
The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n–1 are in the ratio ______.
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.
