Question

What conjecture can you make about the sum of the first 20 positive odd numbers?

Answer 1

Here's the idea (not necessarily the correct idea!) that first comes to mind:

If we estimate the average of the first 20 positive numbers, we get approx. 10. So, imagine having 20 - 10s and adding them together; that'd give us the approx. sum (20)(10), or 200, of the first 20 integers.

Again: this is an ESTIMATE only.

There is a summation formula for consecutive integers:\[\sum_{i=1}^{n}i=\frac{ n(n+1) }{ 2 }\]
If we substitute n=20 (representing the first 20 integers), what will the sum of those 1st 20 integers be? Note that this result, if obtained correctly, will the exact sum of the first 20 integers.

Answer 2

Now compare our approximate sum (200) with the exact sum ( ?? ). Pretty close, eh?