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1997 PMWC Problems/Problem T2

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Problem

Evaluate

\begin{eqnarray*}&& 1 \left(\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7...

Solution

We can group them:

\dfrac{1}{1}+\dfrac{1+3}{2}+\dfrac{1+3+5}{3}+\cdots+\dfrac{1+3+5+7+9+\ldots+19}{10}

The sum of the first n odd numbers is n^2, so we can simplify:

\dfrac{1}{1}+\dfrac{4}{2}+\dfrac{9}{3}+\cdots+\dfrac{100}{10}

That's just 1+2+3+4+5+6+7+8+9+10=\frac{10(10+1)}{2}=55.

See also

1997 PMWC (Problems)
Preceded by
Problem T1
Followed by
Problem T3
I: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T: 1 2 3 4 5 6 7 8 9 10
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