Write the sum using sigma notation. $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }$
A) $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n + 1 ) }$
B) $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n - 1 ) }$
C) $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n + 1 ) }$
D) $\sum _ { n = 1 } ^ { 1001 } \frac { 1 } { n ( n + 1 ) }$
E) $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n - 1 ) }$

Question

Quizplus · Accepted Answer

The sum starts at 1 and goes up to 999, so the summation should start at n=1 and end at n=999. The terms in the sum are of the form $\frac {1}{n(n+1)}$, which matches $\frac {A}{n} + \frac {B}{n+1}$. Solving for $A$ and $B$ gives $A=1$ and $B=-1$, so the sum can be written as $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n + 1 ) }$.

Write the Sum Using Sigma Notation 11⋅2+12⋅3+13⋅4+…+1999⋅1000\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }1⋅21​+2⋅31​+3⋅41​+…+999⋅10001​

Write the Sum Using Sigma Notation $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }$