Telescoping Series 5 – Introducing The Telescope

Algebra Level 1

Given the following, evaluate 1 1 × 2 + 1 2 × 3 + 1 3 × 4 \frac{1}{1\times2} + \frac{1}{2\times3} + \frac{1}{3\times4} :

1 1 × 2 = 1 1 − 1 2 1 2 × 3 = + 1 2 − 1 3 + 1 3 × 4 = + 1 3 − 1 4 \begin{array}{lllll} & \frac{1}{1\times2}= & \frac{1}{1} & - \frac{1}{2} & &\\ & \frac{1}{2\times3}= & &+ \frac{1}{2} & - \frac{1}{3} & \\ + & \frac{1}{3\times4}=& & & + \frac{1}{3} & - \frac{1}{4} \\ \hline \end{array}

3 4 \frac{3}{4} 1 2 \frac{1}{2} 9 10 \frac{9}{10} 2 3 \frac{2}{3}

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Arron Kau by Arron Kau

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8 solutions

Krishna Garg
Apr 10, 2014

Summing up gives 1 -1/4 =3/4 Ans K.K.GARG,India

5 Helpful 0 Interesting 0 Brilliant 0 Confused

eassy

Hameer Rai - 7 years, 2 months ago

1/2+1/6+1/12=(6+2+1)/12 =9/12 =3/4

Mukhtar Ahmad - 7 years, 2 months ago

yo yo yo, i solved it

Armaan Das - 7 years, 1 month ago

so simple nothing to expplain i tnk

Abbas Khan - 7 years, 1 month ago

Simple

Pilol Kumang - 7 years, 1 month ago
Aravind Suresh
Apr 10, 2014

Canceling pairs are (-1/2 ,1/2) (-1/3 ,1/3). Only left out thing is 1-(1/4)=(3/4).

2 Helpful 0 Interesting 0 Brilliant 0 Confused

The problem is solve by fraction method bh using these steps; (1/1X2) is first value, (1/2X3) is second value and (1/3X4) is third value. We solve it by; 

       =(1/2)+(1/6)+(1/12)
Now, take L.C.M;
       =(1+2+6)/12
       =9/12
 Divide Numenator and Denomenator by 3;
       =3/4 (is the answer)
1 Helpful 0 Interesting 0 Brilliant 0 Confused

1-1/2+1/2-1/3+1/3-1/4= 1-1/4=3/4

0 Helpful 0 Interesting 0 Brilliant 0 Confused

We can calculate it by 1/2 +1/6+1/12

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Nikko Quirap
Apr 12, 2014

Only add 1 and -1/4

1 - 1/4 thus 3/4

0 Helpful 0 Interesting 0 Brilliant 0 Confused

1/2+1/6+1/12=3/4 which implies answer is 1-1/4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Abid Ali
Apr 10, 2014

(1/2)+(1/6)+(1/12)=3/4

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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