Algebra

Partial Fractions

Partial Fractions: Level 2 Challenges

         

n=11n2+3n+2= ?\large \displaystyle \sum_{n = 1}^{\infty} \dfrac {1}{n^2 + 3n + 2} = \ ?

1+12+16+112+120+= ?1 + \frac {1}{\color{#3D99F6}2} + \frac {1}{\color{#3D99F6}6} + \frac {1}{\color{#3D99F6}{12}} + \frac {1}{\color{#3D99F6}{20}} + \ldots = \ \color{teal}?

Consider the following pattern:

11×2=111212×3=121313×4=1314\begin{aligned} \frac{1}{1\times 2} & = \frac{1}{1} - \frac{1}{2} \\ & \\ \frac{1}{2\times 3} & = \frac{1}{2} - \frac{1}{3} \\ & \\ \frac{1}{3\times 4} & = \frac{1}{3} - \frac{1}{4} \\ \vdots & \end{aligned}

Following the pattern above, if 111×12=1a1b, \displaystyle \frac{1}{11\times 12} = \frac{1}{a} - \frac{1}{b}, what are the values of a a and b b?

11×2+12×3+13×4++199×100= ? \large \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\cdots +\frac{1}{99 \times 100} = \ ?

113+135+157+179+= ?\large \frac{1}{1\cdot 3} + \frac{1}{3\cdot 5} +\frac{1}{5\cdot 7} + \frac{1}{7 \cdot 9} + \cdots = \ ?

×

Problem Loading...

Note Loading...

Set Loading...