If the second differences are a constant = 5, then the series of first differences is

9 14 19 24 29 34 ...

If the first series number is 4, then the series of interest is

4, (4+9)=13, (13+14)=27, (27+19)=46, (46+24)=70, ...

Your series is

4, 13, 27, 46, 70, ...

A series with a first term (N=1) of x

x

As you can see, this describes a polynomial series. An arithmetic series will have d2=0. A geometric series will have an infinite number of non-zero Nth differences.

Your series has x

x

Then

x

= 4 + 36 + 30 = 70

9 14 19 24 29 34 ...

If the first series number is 4, then the series of interest is

4, (4+9)=13, (13+14)=27, (27+19)=46, (46+24)=70, ...

Your series is

4, 13, 27, 46, 70, ...

A series with a first term (N=1) of x

_{1}, initial first difference of d1, initial second difference of d2, initial 3rd difference of d3, etc. will have this equation for the Nth term:x

_{N}= x_{1}+ d1(N-1) + d2(N-1)(N-2)/2 + d3(N-1)(N-2)(N-3)/(2*3)As you can see, this describes a polynomial series. An arithmetic series will have d2=0. A geometric series will have an infinite number of non-zero Nth differences.

Your series has x

_{1}=4, d1=9, d2=5, d3=0, so the Nth value in the series isx

_{N}=4 + 9(N-1) + 5(N-1)(N-2)/2 = 4 + (N-1)(9 + 5(N-2)/2)Then

x

_{5}= 4 + 9(5-1) + 5(5-1)(5-2)/2 = 4 + 9(4) + 5(4)(3)/2= 4 + 36 + 30 = 70