Question

10. Sum to n terms the following series :
1+ (1+1/2)+ (1+1/2+1/4)....​

Answer 1

Answer:

LetT

n

be then

th

term of the given series,

Then,

T

n

= 1+ 2+ 3+ .....+ n

=

2

n

[2× 1+ (n− 1)× 1]

=

2

n

[2+ n− 1]=

2

n

(n+ 1)

=

2

n 2

+

2

n

Let

S

n

denoted sum of n terms of the given series

S

n

K=1

∑

nT

K

=

K=1

∑

n

[

2

K 2

+K/2]

S

n

2

1

K=1

∑

n

K 2

2

1

K=1

∑

nK

S

n

2

1

[

6

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

]+

2

1

[

2

n(n+1)

]

S

n

12

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

S

n

12

n(n+1)[2n+4]

S

n

12

n(n+1)

× 2(n+ 2)

S

n

6

n

(n+ 1)(n+ 2)