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 x1, initial first difference of d1, initial second difference of d2, initial 3rd difference of d3, etc. will have this equation for the Nth term:
xN = x1 + 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 x1=4, d1=9, d2=5, d3=0, so the Nth value in the series is
xN=4 + 9(N-1) + 5(N-1)(N-2)/2 = 4 + (N-1)(9 + 5(N-2)/2)
Then
x5 = 4 + 9(5-1) + 5(5-1)(5-2)/2 = 4 + 9(4) + 5(4)(3)/2
= 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 x1, initial first difference of d1, initial second difference of d2, initial 3rd difference of d3, etc. will have this equation for the Nth term:
xN = x1 + 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 x1=4, d1=9, d2=5, d3=0, so the Nth value in the series is
xN=4 + 9(N-1) + 5(N-1)(N-2)/2 = 4 + (N-1)(9 + 5(N-2)/2)
Then
x5 = 4 + 9(5-1) + 5(5-1)(5-2)/2 = 4 + 9(4) + 5(4)(3)/2
= 4 + 36 + 30 = 70