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Analysis of AlgorithmsAsymptotic AnalysisWorst, Average and Best CasesAsymptotic NotationsLittle o and little omega notationsLower and Upper Bound TheoryAnalysis of LoopsSolving RecurrencesAmortized AnalysisWhat does 'Space Complexity' mean ?Pseudo-polynomial AlgorithmsPolynomial Time Approximation SchemeA Time Complexity Question
Asymptotic Analysis
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Lower and Upper Bound Theory
Analysis of Loops
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Polynomial Time Approximation Scheme
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Searching Algorithms
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Mathematics | Some theorems on Nested Quantifiers
Mathematics | Rules of Inference
PDNF and PCNF in Discrete Mathematics
Mathematics | Introduction of Set theory
Mathematics | Set Operations (Set theory)
Inclusion-Exclusion and its various Applications
Mathematics | Power Set and its Properties
Mathematics | Partial Orders and Lattices
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Discrete Mathematics | Representing Relations
Mathematics | Representations of Matrices and Graphs in Relations
Mathematics | Closure of Relations and Equivalence Relations
Number of possible Equivalence Relations on a finite set
Mathematics | Classes (Injective, surjective, Bijective) of Functions
Mathematics | Total number of possible functions
Discrete Maths | Generating Functions-Introduction and Prerequisites
Mathematics | Generating Functions – Set 2
Mathematics | Sequence, Series and Summations
Mathematics | Independent Sets, Covering and Matching
Mathematics | Introduction to Proofs
Discrete Mathematics | Hasse Diagrams
Introduction to Mojette transform
Mathematics | Rings, Integral domains and Fields
Mathematics | The Pigeonhole Principle
Mathematics | Combinatorics Basics
Mathematics | PnC and Binomial Coefficients
Mathematics | Generalized PnC Set 1
Mathematics | Generalized PnC Set 2
Corollaries of Binomial Theorem
Number of triangles in a plane if no more than two points are collinear
Mathematics | Sum of squares of even and odd natural numbers
Finding nth term of any Polynomial Sequence
Discrete Mathematics | Types of Recurrence Relations – Set 2
Mathematics | Graph Theory Basics – Set 1
Mathematics | Graph Theory Basics – Set 2
Graph Types and Applications
Mathematics | Euler and Hamiltonian Paths
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Betweenness Centrality (Centrality Measure)
Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph
Graph measurements: length, distance, diameter, eccentricity, radius, center
Relationship between number of nodes and height of binary tree
Mathematics | Matrix Introduction
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Doolittle Algorithm : LU Decomposition
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Easy
Given a few terms of a sequence, we are often asked to find the expression for the nth term of this sequence. While there is a multitude of ways to do this, In this article, we discuss an algorithmic approach which will give the correct answer for any polynomial expression. Note that this method fails for trigonometric, exponential or other transcendental sequences.
We will take a few examples and discuss the method as we solve these examples.
Example 1
Let us take the sequence
S = 3, 9, 15, 21, .... 

Trivial Solution
We can easily see that this is an A.P series. The starting term is 3 and the common difference is 6. Hence the nth term of the sequence is 3 + (n-1) * 6.
Our Solution
Since we know that an nth order polynomial has the form:
Polynomial = Pn = a0 + a1*x^1 + a2*x^2 + ... + an*x^n

Since we already know a few terms of the sequence, we can substitute these values in the above expression and obtain some equations. Then we need to solve this system of linear equations and obtain the coefficients a0, a1, ... an.
The trick is to somehow know the order of the polynomial.
This is simple enough to do if we remember that the nth order forward difference is constant for an nth order polynomial. Also all higher order forward differences are 0. Let us obtain the forward differences for the series
3   9   15   21   27 ...
  6   6    6     6

Since the first forward difference is constant, we conclude that the sequence is a first order polynomial.
1st order polynomial Sn  = a0 + a1*n
For n = 1, Sn = 3
3 = a0 + a1   .... (1)
For n = 2, Sn = 9 
9 = a0 + 2*a1  .... (2)

Solving (1) and (2) we get
a1 = 6 and a0 = -3

Hence we obtain Sn = -3 + 6*n

Note that we obtain slightly different formulations of the sequence in the above two methods. But both these formulations are correct and can easily be converted from one form to another
Example 2
Lets us take the sequence S =
 9   24    47   78   117  164  219 ...
   15   23   31   39   47    55 
      8    8    8    8    8

Since the 2n order forward difference is constant, the Sequence is a 2nd order
polynomial

S = a0 + a1*n + a2 *n*n
Substituting n = 1, 2, 3 and corresponding entries we get
9 = a0 + a1 + a2 ... (1)
24 = a0 + 2*a1 + 4*a2 ... (2)
47 = a0 + 3*a1 + 9*a3 ... (3)

Writing matrices for above equations
A = 1 1 1  
    1 2 4
    1 3 9
B = 9 24 47
X = [a0 a1 a2] = inv(A)*B = [ 2 3 4]
Hence a0 = 2, a1 = 3, a2 = 4
Therefore S = 2 + 3*n + 4 * n * n

Example 3
S = 28, 168, 516, 1168, 2220 ...

Forward Differences
28   168   516   1168    2220
  140   348   652    1052
     208   304    400
         96     96

The Sequence is a third order polynomial
S = a0 + a1*n + a2*n*n + a3*n*n*n
Substituting n = 1, 2, 3, 4 and corresponding values
28 = a0 + a1 + a2 + a3
168 = a0 + 2*a1 + 4*a2*a2 + 8*a3*a3*a3
516 = a0 + 3*a1 + 9*a2*a2 + 27*a3*a3*a3
1168 = a0 + 4*a1 + 16*a2*a2 + 64*a3*a3*a3  

Writing matrices for above equations
A = 1 1 1 1
    1 2 4 8
    1 3 9 27
    1 4 16 64 
B = 28, 168, 516, 1168
X = [a0 a1 a2 a3] = inv(A)*B = [0, 4, 8, 16]

Therefore S = 4*n + 8*n*n + 16*n*n*n

Note: To solve an nth order Sequence we need to solve a system of n equations. These equations can either be solved by hand or by using matrix-based math calculators like MATLAB, Octave etc. A free online version of Octave is available at Octave Online.
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