How to find the coefficient of x^50 in the expansion of the following?
(1+x)^1000 + x(1+x)^999 + x^2(1+x)^998 + . . . . +x^1000
Options are:
A) "^1000C_50
B) "^1001C_50
C) "^1002C_50
D) None of the above
My work so far:
Since we want the term x^50 only here, we get the term only till the following part in expansion given to us:
(1+x)^1000 + x(1+x)^999 + . . . . +(1+x)^950x^50
Now expanding the terms to single out those containing x^50 using Binomial expansion I will get:
"^1000C_50x^50 + "^999C_49x^50 + "^998C_48x^50 + . . . . + "^950C_0x^50
rArr Sum of coefficients = "^1000C_50 + "^999C_49 + . . . +"^950C_0
I know the identity "^nC_r + "^nC_(r+1) = "^(n+1)C_(r+1)
But I couldn't use it to shorten the expansion.
How should I go about this?
Options are:
A)
B)
C)
D) None of the above
My work so far:
Since we want the term
Now expanding the terms to single out those containing
I know the identity
But I couldn't use it to shorten the expansion.
How should I go about this?
1 Answer
Explanation:
This is a GP of 1001 terms, with first term
It is easy to see from this that the desired coefficient( that of