posted by Sandy .
State the rule for this pattern:
What about: subtract the last number and add 2 to the next term
using your rule, do you get the given sequence?
Here's a method of attack:
You know that 1+3+5+...+n = n^2
adding up the 1st n odd numbers gives you n^2.
In your sequence, you have
1 1+4 1+4+6 1+4+6+8
which looks very similar. So, expect something with n^2 in it.
1 = 2^2-3
5 = 3^2-4
11 = 4^2-5
19 = 5^2-6
So, it looks to me like
Tn = (n+1)^2 - (n+2) = n^2+n-1