Telescopic Series (With An Easy Pattern!)

Level 2

1 2 \frac{1}{2} + 1 6 \frac{1}{6} + 1 12 \frac{1}{12} + 1 20 \frac{1}{20} +...+ 1 10100 \frac{1}{10100}

Optional extension: Use summation to calculate the answer.

100/101 0.99 0.988 868/888

Xin Ze Cai by Xin Ze Cai , 16, Australia

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1 solution

Xin Ze Cai
Sep 1, 2019

Notice the following pattern:

1 2 \frac{1}{2} + 1 6 \frac{1}{6} = 4 6 \frac{4}{6} .

4 = 2 x 2. The last fraction was the second term in the series.

1 2 \frac{1}{2} + 1 6 \frac{1}{6} = 4 6 \frac{4}{6} + 1 12 \frac{1}{12} = 9 12 \frac{9}{12} .

9 = 3 x 3. The last fraction was the third term in the series.

1 2 \frac{1}{2} + 1 6 \frac{1}{6} = 4 6 \frac{4}{6} + 1 12 \frac{1}{12} + 1 20 \frac{1}{20} = 16 20 \frac{16}{20} .

16 = 4 x 4. The last fraction was the fourth term in the series.

Applying the same principles to 10100, we get: 10 0 2 10100 \frac{100^2}{10100} = 10000 10100 \frac{10000}{10100} = 100 101 \frac{100}{101} .

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