what is the coefficient of x¹¹ in the product (1-x)(1-4x)(1-4²x) (1-4¹¹x) - Maths - Binomial Theorem

Question

what is the coefficient of x¹¹ in the product
(1-x)(1-4x)(1-4²x) (1-4¹¹x) ?

Ishwarmani · Accepted Answer

Dear
Student,
Please find
below the solution to the asked query:

The coefficient of x11 in 1-x1-4x1-42x
1-411x=-1-4-42 -410+-1-4-42 -49-411+-1-4-42 -48 -410-411+-1-4-42
-47 -49-410-411+ +-4-42 -411=-1-4-42 -410-411-411+-1-4-42
-49-410-411-410+-1-4-42 -48-49-410-411-49+-1-4-42
-47-48-49-410-411-48+ +-4-42 -411-1=-1-4-42
-410-4111-411+1-410+1-49+1-48+ +1-1=-14×42×
×4111-411+1-410+1-49+1-48+ +1-1=-41+2+
+111-411-412-1-4-1                            As, the values inside 
 are in GP with a=1-411,r=-4 and n=12=-41111+12412-1-5-411                                                 As, sum of first n narural numbers=nn+12=-41111+12-5-411412-1=-411×65×411412-1=-466-115412-1=-455412-15

Hope
this
information will clear
your doubts about the
topic
Regards

what is the coefficient of x¹¹ in the product (1-x)(1-4x)(1-4²x).....(1-4¹¹x) ?