Friday, March 15, 2013

Finding the n th term in a sequence

We encounter situations to find the nth term of a series that is neither in arithmetic progression nor in geometric progression, and here we use the below simple technique to figure out the nth term.

Say we have a sequence in which the nth term is to be figured out. If the sequence of numbers are in Arithmetic progression or Geometric progression or Harmonic progression we have direct formula to find the     nth term. But if the series follows none of the above progressions, even then we have a simple method to calculate the nth term applying basic mathematical tools.

Say the sequence is 

10  12  16  22  30  . . . . .  

We shall consider the general expression aN+ bN + c 

Where N denotes the term number and a,b,c are constants, now we can generate equations using the general expression as shown below:

Equation 1 : Put N = 1 in general expression and equate it to 10 ( first term )          a+ b +c = 10
Equation 2 : Put N = 2 in general expression and equate it to 12 ( second term )   4a+2b+c = 12
Equation 3 : Put N = 3 in general expression and equate it to 16 ( third term )       9a+3b+c = 16

Now solving the above three equations we obtain the values of constants a,b and c.
Solving of equations may be done using matrices or linear equation formula. 

By elimination method Eliminating c in Equation 1 and 2 (Subtracting eq 2 from eq 1) we get  3a + b = 2
Again Eliminating c in Equation 2 and 3 (subtracting eq 2 from 3) we get 5a +b = 4

Now solving the equations thus obtained, 3a+b= 2 and 5a+b=4.

Eliminating b from below equations ...
5a + b = 4
3a + b = 2 

We get 2a = 2 and a =1 from the above equations hence finding b value from the same equation b = -1 
and similarly c = 10

The general expression becomes N-N + 10
and thats the nth term of the given series. 
The same way one can find the nth term for any series.