Question

Plz help me plz plz plz

Answer 1

Answer:

Step-by-step explanation:

Alpha Centauri is a triple star system located just over four light years, or about 25 trillion miles, from Earth. While this is a large distance in terrestrial terms, it is three times closer than the next nearest Sun-like star.

Answer 2

Solution

By partial fraction decomposition,

$\dfrac{1}{AB} =\dfrac{1}{B-A} (\dfrac{1}{A} -\dfrac{1}{B} )$ …①

Here, $A=n$ and $B=n+1$.

$\implies B-A=1$ …②

We can simplify this and rewrite the series.

$\implies \dfrac{1}{AB} =\dfrac{1}{A} -\dfrac{1}{B}$ (∵①, ②)

We can see the problem is a telescoping series, using this method.

Given series,

$\implies\displaystyle\sum^{1000}_{k=1}(\dfrac{1}{n} -\dfrac{1}{n+1} ) =1-\dfrac{1}{2} +\dfrac{1}{2} -\dfrac{1}{3} +...+\dfrac{1}{1000} -\dfrac{1}{1001}$

$\implies\displaystyle\sum^{1000}_{k=1}(\dfrac{1}{n} -\dfrac{1}{n+1} ) =1 -\dfrac{1}{1001}$

$\implies\displaystyle\sum^{1000}_{k=1}(\dfrac{1}{n} -\dfrac{1}{n+1} ) =\dfrac{1000}{1001}$

Thus, the correct option is $(B)$.

Learn More

About partial fraction decomposition.

• How is ① derived?

We start from $\dfrac{1}{B} -\dfrac{1}{A}$,

$\implies \dfrac{1}{B} -\dfrac{1}{A}=\dfrac{A-B}{AB}$

Dividing both sides,

$\implies \dfrac{1}{A-B} (\dfrac{1}{B} -\dfrac{1}{A} )=\dfrac{1}{AB}$

Simplifying further,

$\implies \dfrac{1}{AB} =\dfrac{1}{B-A} (\dfrac{1}{A} -\dfrac{1}{B} )$

Hence derived.