Pentagonal Numbers - Generalized Step

The formula for the number of dots that will be added to the nth figure is $3n-2$ (from the earlier problem "Pentagonal Numbers - How To Proceed?"). Substituting $n$ with $n+1$ , we get $3(n+1)-2 \Rightarrow 3n+3-2 \Rightarrow \boxed{3n+1}$

The general formula for the number of dots in the $n$ th pentagonal number is $\frac {3n^{2}-n}{2}$ .

So the answer to the problem would be

no. of dots in $n+1$ th number - no. of dots in $n$ th number = $\frac {3(n+1)^{2}-(n+1)}{2}$ $-$ $\frac {3n^{2}-n}{2}$ . = $\boxed{3n + 1}$

Use arithmetic progressions to find out nth and (n+1)th term. Then take the difference.

If n>=1 is the lenght of the side of the biggest pentagon in the figure, you always sum 3n +1 to the number of dots from the previous figure.

No of dots follows series 1 5 12 22 ? ...so next num will be 35 as dif between consecutive num follows sequence 4(5-1) 7(12-5) 10(22-12) 13(35-22)....so on which follows a sequence in GP a+(n-1)d=4+(n-1)3=3n+1...so num of dots increased in next fig is nothing but difference which is equal to 3n+1