Question

show that c0-3c1+7c2-10c3=0​

Answer 1

Answer:

2n.(n+1)

Step-by-step explanation:

Consider the following

(1+x

2

)

n

=1+

n

C

1

x

2

+

n

C

2

x

4

+

n

C

3

x

6

...

n

C

n

x

2n

Multiplying both sides with x, we get

x(1+x

2

)

n

=x+

n

C

1

x

3

+

n

C

2

x

5

+

n

C

3

x

7

...

n

C

n

x

2n+1

Differentiating both sides with respect to x, we get

[(1+x

2

)

n

+nx(2x)(1+x

2

)

n−1

]

x=1

=1+3

n

C

1

+5

n

C

2

+...(2n+1)

n

C

n

Hence

1+3

n

C

1

+5

n

C

2

+...(2n+1)

n

C

n

=[(1+x

2

)

n

+nx(2x)(1+x

2

)

n−1

]

x=1

=2

n

+2n(2

n−1

)

=2

n

+n2

n

=2

n

(n+1)