# How to find the coefficient of #x^50# in the expansion of the following?

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#(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