Sometime ago, in one of newsletters I came across a question to find next number in series. The series was 3, 11, 20, 27, 29, 23. The next number in series needed to be found. Well, I tried as much as I can and stopped when my brain gave up on me. I was never too bright at math. I wasn’t bad either but I wasn’t this old either at that time.

So can you help me find answer?

A for effort was given to a colleague who answered this correctly and I can clearly see why. He answered it very thoroughly and I am going to publish it with due credits to him (name not published for privacy).

As he wrote:

“The series given was 3, 11, 20, 27, 29, 23

The solution, the way I saw it was to find some function, f(x) that would equal:

f(1) = 3, f(2) = 11, f(3) = 20, etc

so that I could predict the value of f(7).

I found the differences between each set of answers to go down 3 levels, with a shared difference of -3. Three levels means I need to find a third degree polynomial f(x) = ax^3 + bx^2 + cx + d.

I then needed to find the values for a, b, c and d. There are 4 unknowns, so I need 4 equations.

The four equations are:

f(1) = a(1)^3 + b(1)^2 + c(1) + d = 3

f(2) = a(2)^3 + b(2)^2 + c(2) + d = 11

f(3) = a(3)^3 + b(3)^2 + c(3) + d = 20

f(4) = a(4)^3 + b(4)^2 + c(4) + d = 27

Using linear algebra and putting these into a matrix and reducing to echelon form, I solved for a, b, c and d.

a=-1/2, b=7/2, c=1, d=-1

That makes f(x) = -1/2 x^3 + 7/2 x^2 + x – 1

Calculating out f(x):

X | -1/2 x^3 | 7/2 x^2 | X | -1 | f(x) |

0 | 0 | 0 | 0 | -1 | -1 |

1 | -0.5 | 3.5 | 1 | -1 | 3 |

2 | -4 | 14 | 2 | -1 | 11 |

3 | -13.5 | 31.5 | 3 | -1 | 20 |

4 | -32 | 56 | 4 | -1 | 27 |

5 | -62.5 | 87.5 | 5 | -1 | 29 |

6 | -108 | 126 | 6 | -1 | 23 |

7 | -171.5 | 171.5 | 7 | -1 | 6 |

8 | -256 | 224 | 8 | -1 | -25 |

9 | -364.5 | 283.5 | 9 | -1 | -73 |

Finally, the answer to the question is 6, highlighted above in red.”

Issshh… I knew a bit of math. Now i am lost again :). not because something is wrong here… because, i fell into a wrong trap of using mind :).

Great job though!