Telescoping Series 6 – Putting It Together

Algebra Level 1

Evaluate:

1 1 × 2 + 1 2 × 3 + 1 3 × 4 + ⋯ ⋯ ⋯ + 1 99 × 100 \frac{1}{1\times 2} + \frac{1}{2\times 3} + \frac{1}{3\times 4} + \cdots \; \cdots \; \cdots + \frac{1}{99\times 100}


The answer is 0.99.

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Arron Kau by Arron Kau

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7 solutions

Yogesh Ghadge
Apr 10, 2014

1/1*2 will be split as 1-1/2 If we do it simultaneously we get 1-1/2 plus 1/2-1/3 We will get alternative plus and minus no of same value They will get cancelled and the remaining no will be 1-1/100 we get 99/100 so the answer is 0.99

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

the last term is1/(99*100), so 99/100=.99

3 Helpful 0 Interesting 0 Brilliant 1 Confused
Krishna Garg
Apr 9, 2014

Since all units are getting cancelled by + and - values except 1 and 1/99X100.therefore sum of remaining items will be 1 plus 1/99 -1/100 and that is 1.000101 Ans.

K.K.GARG,India

3 Helpful 0 Interesting 0 Brilliant 0 Confused

1/1 2+1/2 3+1/3*4.......1/(n)(n+1) = n/(n+1) (where n is the no. of terms) since there are 99 terms the answer is 99/(99+1) which is = 99/100=0.99

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Maham Zaidi
Apr 9, 2014

According to the pattern, the previous problem showed the answer 3/4 when all the sums were added. 3/4 were last fraction's 3 x 4. Therefore, the last fraction here is 1/(99x100) and the answer is 99/100; which is 0.99 in decimal or 1.0 if you estimate it.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Bao Tran Tong
May 12, 2014

we have: 1/[n.(n+1)]= 1/n- 1/(n+1) => 1/(1.2) + 1/(2.3) + ....+ 1/(99.100) = 1-1/2+1/2-1/3+...+1/99-1/100=1-1/100= 99/100

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Isabella Amabel
Apr 20, 2014

100/101 = 0.990099009901

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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