A sequence is called an arithmetic progression of the �rst order if the
di�erences of the successive terms are constant. It is called an arith-
metic progression of the second order if the di�erences of the successive
terms form an arithmetic progression of the �rst order. In general, for
k � 2, a sequence is called an arithmetic progression of the k-th order
if the di�erences of the successive terms form an arithmetic progression
of the (k 1)-th order.
The numbers
4; 6; 13; 27; 50; 84
are the �rst six terms of an arithmetic progression of some order. What
is its least possible order? Find a formula for the n-th term of this
progression.
The numbers 4; 6; 13; 27; 50; 84 do not form an arithmetic progression as the differences between succsessive terms are not constant.
If you take the successive differences of the terms given,
n.......1....2....3....4....5....6...
N.......4....6...13...27...50...84...
1st Diff..2....7...14....23...34...
2nd Diff....5....7....9.....11
3d Diff.......2....2.....2
you find that the 3rd differences are constant meaning that the sequence is a finite difference sequence of the 3rd order, the defining expression being of the form N = an^3 + bn^2 + cn + d.
Using the given data,
a(1)^3 + b(1)^2 + c(1) + 1 = 4 or a + b + c + d = 4
a(2)^3 + b(2)^2 + c(2) + 2 = 6 or 8a + 4b + 2c + 2 = 6
a(3)^3 + b(3)^2 + c(3) + 3 = 13 or 27a + 9b + 3c + 3 = 13
a(4)3 + b(4)^2 + c(4) + 4 = 27 or 64a + 16b + 4c + 4 = 27
Solve for a, b, c and d and substitute back into N = an^3 + bn^2 +cn + d
To determine the least possible order of the arithmetic progression and find a formula for the nth term, we need to observe the differences between successive terms.
Given the sequence: 4, 6, 13, 27, 50, 84
Let's calculate the differences:
2, 7, 14, 23, 34
From the differences, we can see that they do not form an arithmetic progression. To determine it, we need to find the differences between these differences:
5, 7, 9, 11
Now, we can see that these differences (5, 7, 9, 11) form an arithmetic progression of the first order.
Since the differences between differences form an arithmetic progression of the first order, we conclude that the given sequence is an arithmetic progression of the second order.
To find the formula for the nth term in this arithmetic progression, we need to find a pattern for the differences between the terms.
First, let's list the differences again: 2, 7, 14, 23, 34
We can see that the differences increase by adding consecutive odd numbers:
2 + 5 = 7, 7 + 7 = 14, 14 + 9 = 23, 23 + 11 = 34
Thus, the nth difference can be represented by the formula: d(n) = d(n-1) + (2n-1)
Now, let's apply this formula to find the differences:
d(1) = 2
d(2) = d(1) + (2 * 2 - 1) = 2 + 3 = 5
d(3) = d(2) + (2 * 3 - 1) = 5 + 5 = 10
d(4) = d(3) + (2 * 4 - 1) = 10 + 7 = 17
d(5) = d(4) + (2 * 5 - 1) = 17 + 9 = 26
The differences between the terms are: 2, 5, 10, 17, 26
Now, let's calculate the terms using the differences and the given sequence:
Term 1: 4
Term 2: 4 + 2 = 6
Term 3: 6 + 5 = 11
Term 4: 11 + 10 = 21
Term 5: 21 + 17 = 38
Term 6: 38 + 26 = 64
Thus, the formula for the nth term in this arithmetic progression of the second order is:
a(n) = 4 + (n-1)*2 + 2*(n-1)*(n-2)
Note: The formula is derived based on the given terms and their respective positions in the sequence.