# If The Second Differences Of A Sequence Are A Constant Of 5, The First Of The First Differences Is 9, And Their First Term Is 4, Which Are The First Five Terms Of Sequence?

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, ...

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
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If the second difference of a sequence are a constant 2, the first of the first differences is 10, and the first term is 5, which are the first terms of the sequence?
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The first term of the sequence is 4
The difference is between first two term is 9

second term 4+9=13

after second the difference is constant 5

Third term = 13 +5 =18

Fouth term = 18 + 5 =23

Fifth term = 23+5 = 28

The numbers are 4, 13, 18, 23,28 >>Answer
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