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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Lower Bound

Stage: 3 Challenge Level: Challenge Level:1

Investigate the following sequence of fraction sums:
\begin{eqnarray} \frac{1}{2} &+& \frac{2}{1} = \\ \frac{2}{3} &+& \frac{3}{2} = \\ \frac{3}{4} &+& \frac{4}{3} = \\ \frac{4}{5} &+& \frac{5}{4} = \end{eqnarray}
What would you get if you continued this sequence for ever?

What do you think will happen if you add the squares of these fractions, that is:
\begin{eqnarray} \left(\frac{1}{2}\right)^2 &+& \left(\frac{2}{1}\right)^2 = \\ \left(\frac{2}{3}\right)^2 &+& \left(\frac{3}{2}\right)^2 = \end{eqnarray}
and so on?