If (x-1)^100 = a0+a1x1+a2x2+a3x3...a100x100 then find the value of a0
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Step-by-step explanation:
Here a
i
=
10
C
i
ie,(a
0
−a
2
+a
4
−a
6
+a
8
−a
10
)
2
+(a
1
−a
3
+a
5
−a
7
+a
9
)
2
=(
10
C
0
−
10
C
2
+
10
C
4
−
10
C
6
+
10
C
8
−
10
C
10
)
2
+(
10
C
1
−
10
C
3
+
10
C
5
−
10
C
7
+
10
C
9
)
2
⇒((
10
C
0
−
10
C
10
)+(
10
C
8
−
10
C
2
)+(
10
C
4
−
10
C
6
))
2
+((−1)(−2)
2
10
)
2
=2
10
Since ∑
i=0
[n/2]
n
C
2i
=
⎩
⎪
⎨
⎪
⎧
0,If
4
n+2
∈Integers
(−1)
[(n+2)/4]
2
[n/2]
and ∑
i=0
[n/2]
n
C
2i+1
={
0,If
4
n
∈Integers
(−1)
[n/4]
2
[n/2]
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