Question

if sigma r=1ton tr = n(n+1)(n+2) /12​

Answer 1

Step-by-step explanation:

Let S

n

=∑

r=1

n

t

r

ie, t

n

=S

n

−S

n−1

=

8

n(n+1)(n+2)(n+3)

−

8

(n−1)n(n+1)(n+2)

=

8

n(n+1)(n+2)(n+3−(n−1))

=

2

n(n+1)(n+2)

Let,

t

n

1

=

n(n+1)(n+2)

2

=

n

A

+

n+1

B

+

n+2

C

⇒

n

A

+

n+1

B

+

n+2

C

=

n(n+1)

A(n+1)+Bn

+

n+2

C

=

n(n+1)(n+2)

((A+B)n+A)(n+2)+Cn(n+1)

⇒

n(n+1)(n+2)

(A+B+C)n

2

+(3A+2B+C)n+2A

=

n(n+1)(n+2)

2

ie,A=1,B=(−2),C=1

∴∑

r=1

n

t

r

1

=∑

r=1

n

(

n

1

−

n+1

2

+

n+2

1

)=(

1

1

+

−

2

2

+

2

1

+

+

3

1

−

3

2

++

3

1

+

+

4

1

−

4

2

+

4

1

+

...

+

+

n

1

−

n

2

+

n

1

+

+

n+1

1

−

n+1

2

+

n+2

1

)

⇒∑

r=1

n

t

r

1

=

2

1

−

n+1

1

+

n+2

1

=

2

1

−

(n+1)(n+2)

1