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