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?

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?

Fibs

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:

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?