Q62. If the sum of the series $\frac { 1 } { 1 \cdot ( 1 + \mathrm { d } ) } + \frac { 1 } { ( 1 + \mathrm { d } ) ( 1 + 2 \mathrm {~d} ) } + \ldots + \frac { 1 } { ( 1 + 9 \mathrm {~d} ) ( 1 + 10 \mathrm {~d} ) }$ is equal to 5 , then 50 d is equal to :
(1) 10
(2) 5
(3) 15
(4) 20

Evaluate
$$\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( n \pi + \frac { \pi } { 3 } \right) } { 2 ^ { n } }$$

$$\sum _ { n = 5 } ^ { 14 } \frac { 1 } { 1 + 2 + \cdots + n }$$
What is the value of this sum?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 3 } { 5 }$
D) $\frac { 2 } { 15 }$
E) $\frac { 4 } { 15 }$

$$\left( \sum _ { k = 1 } ^ { 9 } k \right) \cdot \left( \sum _ { n = 1 } ^ { 8 } \frac { 1 } { n ( n + 1 ) } \right)$$
What is the result of this operation?
A) 27
B) 30
C) 32
D) 36
E) 40