Finite and Infinite Sequences

If a set of numbers are such that they follow a specific pattern is called a sequence. If all the terms are added in order then it is called a series. The main types of series and sequences which are taught till intermediate are Arithmetic series and sequences, Geometric Series and Sequences and Harmonic Sequence.

Look at the following sequences:
1. 2, 4, 6, 8, 10, 12,...
2. 3, 9, 27, 81, 243,...
3. 1/2, 1/4, 1/6, 1/8, 1/10, 1/12,...

• Arithmetic sequence: The first sequence (2, 4, 6, 8, 10, 12,...) among the three sequences is an arithmetic sequence. 2+4+6+8+10+12+.... is an arithmetic series. A general arithmetic sequence looks like this.
  a, a+d, a+2d, a+3d,... and its corresponding series is a + a+d + a+2d + a+3d +....
  a is called the first term and d is the common difference. The first term of the example sequence is 2 and common difference is 2.
  
  The n^th term of an A.S. is
  t_n = a + (n-1)d
  and sum of first n terms is
  S_n = (n/2)(2a + (n-1)d)
  n is the number of terms.
  
  How to check whether a sequence is arithmetic sequence.
  
  Difference is found by subtracting any term with its successor term. When all the differences are the same then the sequence is arithmetic sequence and the difference is called common difference.
  


• Geometric sequence: The second series (3, 9, 27, 81, 243,...) among the three sequences is a geometric sequence. 3+9+27+81+243+.... is a geometric series. A general geometric sequence looks like this.
  a, ar, ar², ar³,... and its corresponding series is a + ar + ar² + ar³ + ....
  a is called the first term and r is the common ratio. The first term in the example is 3 and common ratio is 3.
  
  The n^th term of an G.S. is
  t_n = ar^(n-1)
  and sum of first n terms is
  S_n = a(1-rⁿ)/(1-r) if r<1
  else
  S_n = a(rⁿ-1)/(r-1) if r>1
  n is the number of terms.
  
  How to check whether a sequence is geometric sequence.
  
  Ratio is found by dividing any term with its previous term. When all the ratios are the same then the sequence is geometric sequence and ratio is called common ratio.
  


• Harmonic sequence: The third series (1/2, 1/4, 1/6, 1/8, 1/10, 1/12,...) among the three sequences is a harmonic sequence. 1/2+1/4+1/6+1/8+1/10+1/12+.... is a series. A general Harmonic sequence looks like this.
  1/a, 1/(a+d), 1/(a+2d), 1/(a+3d),... and its corresponding series is 1/a + 1/(a+d) + 1/(a+2d) + 1/(a+3d) +....
  
  (1/a) is called the first term. There is no formula to find the sum of n terms of the series.
  
  How to check whether a sequence is harmonic sequence.
  
  Find the reciprocal of all the terms. If the sequence formed is an arithmetic sequence then the sequence is harmonic sequence.