Use the method of successive differences to find the next term.?

Question

Use the method of successive differences to find the next term.
5, 21, 47, 88, 149, 235, …


a) 275
b) 265
c) 240
d) 351

10pts for correct answer

Answer 1

This sequence can be represented as a polynomial. To find the degree of the polynomial, first find the differences between each of the terms. Then repeat the process until such time as all the terms are the same. The number of times you have to find the differences before the terms are all the same is the degree of the polynomial. In this case:

5 21 47 88 149 235
16 26 41 61 86
.. 10 15 20 25
..... 5 .. 5.. 5

We had to find differences three times, so the terms are given by a polynomial of degree three. Now, if we want a closed form solution for the polynomial, we would write ax³+bx²+cx+d=P(x), and substitute in known values for x and P(x) -- e.g. a(1)³ + b(1)² + c(1) + d = 5, a(2)³ + b(2)² + c(2) + d = 21, a(3)³ + b(3)² + c(3) + d = 47, a(4)³ + b(4)² + c(4) + d = 88. This would give us a system of four equations and four unknowns, which we could solve for the closed-form expression. However, we don't actually need the closed-form expression to get the next term -- if we go back to the differences in the terms:

5 21 47 88 149 235
16 26 41 61 86
.. 10 15 20 25
..... 5 .. 5.. 5

We know that the next term in the bottom line will be another 5, so we can compute the next term in the third row will be 25+5, which is 30. So the next term in the second row will be 86+30, which is 116. And finally, the next term in the first row (which is the term we want) is 235+116, which is 351. So the answer is d: 351.

If you're interested, the closed-form expression for the polynomial turns out to be P(x) = 5x³/6 + 61x/6 - 6.

Answer 2

d) 351

5, 21, 47, 88, 149, 235, 351...

The differences between successive terms are:
16, 26, 41, 61, 86...

The differences between these differences are:
10, 15, 20, 25...

So the next difference of differences must be 30.
This results in the next difference of 86+30 = 116.
So the next term is 235+116 = 351

Answer 3

I thought you are a maths lover. You should do it yourself if you love maths.
P.S. I did not know what the method of successive differences was before reading the wonderful solution of Pascal. Thanx Pascal.