find the coefficient of x^98 in
(x-1)(x-2)(x-3)...(x-99)(x-100)?
find the coefficient of x^98 in
(x-1)(x-2)(x-3)...(x-99)(x-100)?
iit jee , 14 Years ago
Grade 12
Algebra> sequence and series...
1 Answers
Ramesh V
Last Activity: 14 Years ago
If we expand (x-1)(x-2).......(x-100) we will get
(x-1)(x-2).......(x-100)=x100 -(1+2+3....100)x99+(Sum of product of all first 100 integers taken two at a time)x98+................
Now we have to find "Sum of product of all first 100 integers taken two at a time" which is the required co-efficient.
(1+2+3+....100)2=12+22+32+.....+1002+2(Sum of product of all first 100 integers taken two at a time)
So Sum of product of all first 100 integers taken two at a time=((1+2+3+....100)2-(12+22+32+.....+1002))/2
Now use 1+2+3+....+n=n(n+1)/2 and 12+22+32+.....+n2=n(n+1)(2n+1)/6
Here n=100
So we get the required co-efficient=12582075
all the best
--
regards
Ramesh
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