The applet below presents you with a sequence of numbers. It's your task to surmise the pattern and guess what would be the next term. Sometimes a pattern may not be unique. This may be very frustrating for which I am sorry. However, this exercise besides being useful and sometimes also enjoyable occupation, affords an opportunity to discuss topics that I consider interesting.

Algorithms used

The applet randomly selects one of the following algorithms. In turn, each of the algorithms has its own random parameters which are mostly coefficients in various formulas.

• Arithmetic series.
  a₀, a₀ + d, a₀ + 2d, a₀ + 3d, ...

• Geometric series.
  a₀, a₀q, a₀q², a₀q³, ...

• Alternate addition and multiplication.
  a₀, a₁ = a₀ + d, a₂ = a₁·q, a₃ = a₂ + d, ...

• Alternate addition of two different quantities.
  a₀, a₁ = a₀ + d_a, a₂ = a₁ + d_b, ...

• Self-descriptive sequence.
  Write a simple number, say, 21. Now scan the number by groups of equal digits at a time. Here we have one "2" and one "1". Thus the next number will be 1211. In this number we have one "1", one "2" and two "1"s. Thus the third number - 111221. The next will be 312211.

• Sum of preceding two.
  a₀ and a₁ are random. a₂ = a₀ + a₁, a₃=a₁ + a₂, ... (If a₀ = 0 and a₁ = 1,, the rseult is the famous Fibonacci sequence.)

• Three-term sum sequence.
  a₀, a₁ and a₂ are random. a₃ = a₀ + a₁ + a₂, ...

• Shifted powers.
  a_n = a₀ + bqⁿ
  Just add a constant to all the members of a geometric series and what a difference!

• Second degree polynomial.
  a_n = an² + bn + c
  

  It's easy to detect this case. Take differences of successive terms:

  Mn = an + 1 - an   = a((n + 1)2 - n2) + b((n + 1) - n) + (c - c)   = a(2n + 1) + b

  So that the differences prove to be the values of a first order polynomial. Taking "second order" differences we get constants:

  Mn - Mn-1 = a((2n + 1) - (2n-1)) + (b-b)   = 2a