If the number of cuts is c then the number of pieces is p=½c²+½c+1.
When c=1, p=2; c=2, p=4; c=3, p=7; c=4, p=11.
Let’s see how we got the formula.
Write the series: 2, 4, 7, 11.
The difference between consecutive terms is 2, 3, 4 and the difference between these is 1, which means that we are adding c pieces to the existing number of pieces with each new cut.
We can write p=Ac²+Bc+C where A, B and C are constant coefficients. The degree is 2 because we had to go to two differences before we found a constant difference.
We need 3 equations to find the three coefficients.
① c=1: A+B+C=2
② c=2: 4A+2B+C=4
③ c=3: 9A+3B+C=7
②-①: 3A+B=2 ④
③-②: 5A+B=3 ⑤
⑤-④: 2A=1, A=½, so B=2-1½=½ and C=2-A-B=1.
Therefore p=½c²+½c+1. Note that when c=4, the formula gives: p=8+2+1=11.