Question

Look at the table. Make a conjecture about the sum of the first 25 positive even numbers.

Answer 1

We notice that the sum is Σ= n(n+1),

Then the SUM of the first 25 even numbersis equal to 25(25+1) = 25 x 26 

Answer 2

Even number are of the form 2n.  The sum of this sequence is the average of the first and last terms times the number of terms.

So the first term is 2 and the 25th term is 50 so the sum is:

(2+50)(25/2)

52*12.5=650

The conjecture they wish you to make is that for n terms, the sum will be:

s=n(n+1)

And for 25 terms that would give you:

s=25*26=650