Posted by **Bhupendra** on Friday, May 11, 2012 at 6:47am.

A sequence is called an arithmetic progression of the rst order if the

dierences of the successive terms are constant. It is called an arith-

metic progression of the second order if the dierences 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 dierences 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.

- Math -
**tchrwill**, Friday, May 11, 2012 at 10:48am
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

## Answer this Question

## Related Questions

Maths - 1..The first 2 terms of a geometric progression are the same as the ...

math - The first, the third and the seventh terms of an increasing arithmetic ...

Math (Geometric Progression) - 5 distinct positive reals form an arithmetic ...

Arithmetic - The first, second and third terms of a geometric progression are 2k...

Math - 5 distinct positive reals form an arithmetic progression. The 1st, 2nd ...

AP Calculus - The 9th term of an arithmetic progression is 4+5p and the sum of ...

arithmetic - Two arithmetic progression have thd same first and last terms.the ...

math - I would need help with example: The sum of three consecutive terms of ...

AP calculus - The consecutive terms of an arithmetic progression are 5-x, 8, 2x...

AP calculus - Given an arithmetic progression -7,-3,1,..., state three ...