Math Formulae: Series

 

• Arithmetic Progression (A.P)
  
  
1. A.P = a₁, a₂, a₃, ...., a_n
   where, a₂ = a + d,
   (or,     a₂ = a - c.d)
   where, d = common difference, and a = first term of A.P
   
   Thus, A.P. = a, a + d, a + 2.d, ...., a + (n - 1).d
   
   
2. n^th term of A.P: a_n = a + (n - 1).d
   where, a = first term of A.P.
   
   
3. Sum of n terms of A.P: S_n = 

   	n
   	2

    [2.a + (n - 1).d]  = 

   	n
   	2

    [a + l]
   where, l = last term (a + (n - 1).d)
   
   
4. Arithmetic Mean: (of n terms)
       A.M. = 

   	(a1 + a2 + a3 + ... + an)
   	n

      = 

   	Sn
   	n

   
   
   
5. Thus, S_n = n × A.M.
   
   
6. If, in a question, we need to assume 3 consecutive terms of A.P, it is convinient to take them as, (a - d), a, (a + d)
   Similarly, if we need to take 5 consecutive terms of A.P, it is convinient to take them as, (a - 2d), (a - d), a, (a + d), (a + 2d)
   
   
7. If, in a question, we need to assume 4 consecutive terms of A.P, it is convinient to take them as, (a - 3d), (a - d), (a + d), (a + 3d)
   where, common difference is 2d.
   
   
8. If a particular number X is added to each term of A.P., then resultant sequence is also in A.P.
   
   
9. If a particular number X is Subtracted from each term of A.P., then resultant sequence is also in A.P.
   
   
10. If a particular number X is multiplied to each term of A.P., then resultant sequence is also in A.P.
    
    
11. If each term of A.P. is divided by X, then resultant sequence is also in A.P, provided X ≠ 0.
    
    
12. If each term of an A.P is added to or Subtracted from corresponding term of another A.P, then resultant sequence is also in A.P, provided that two A.P's have equal number of terms.
    
    
13. If an A.P. (a₁, a₂, ...., a_n) is having odd number of terms, then
         A.M. = 

    	a1 + an
    	2

       = 

    	a2 + an - 1
    	2

       =  ...   = a_k,
    where, a_k = mid-term of A.P,
    and,   k = 

    	n + 1
    	2

    
    
    
14. If an A.P. (a₁, a₂, ...., a_n) is having even number of terms, then
         A.M. = 

    	a1 + an
    	2

       = 

    	a2 + an - 1
    	2

       =  ...   = 

    	ak + ak + 1
    	2

     where, k = 

    	n
    	2

    
    
    

---

• Geometric Progression (G.P)
  
  
1. G.P = a, a.r, a.r², ...., a.r^{(n - 1)}
   where, Common ratio: r = 

   	a2
   	a

      = 

   	a3
   	a2

      =......= 

   	an
   	an - 1

   
   
   
2. n^th term of G.P: a_n = a.r^{(n - 1)}
   where, a = first term of G.P.
   
   
3. Sum of n terms of G.P: S_n = 

   	a.(r n - 1)
   	(r - 1)

      , if r > 1, and
     S_n = 

   	a.(1 - r n)
   	(1 - r)

      , if r < 1
   
   
4. Geometric Mean: (of n terms)
       G.M. = (a₁ . a₂ . a₃ ........ a_n) ^1/n
   
   
5. Thus, S_n = n × A.M.
   
   
6. If, in a question, we need to assume 3 consecutive terms of G.P, it is convinient to take them as, 

   	a
   	r

   ,  a,  a.r
   Similarly, if we need to take 5 consecutive terms of G.P, it is convinient to take them as, 

   	a
   	r 2

   , 

   	a
   	r

   ,  a,  a.r,  a.r ²
   
   
7. If, in a question, we need to assume 4 consecutive terms of G.P, it is convinient to take them as, 

   	a
   	r 3

   , 

   	a
   	r

   ,  a.r,  a.r³ where, common ratio is r².
   
   
8. If a particular number X is multiplied to each term of G.P., then resultant sequence is also in G.P.
   
   
9. If each term of G.P. is divided by X, then resultant sequence is also in G.P, provided X ≠ 0.
   
   
10. If each term of a G.P is multiplied to corresponding term of another G.P, then resultant sequence is also in G.P, provided that two G.P's have equal number of terms.
    
    
11. If a G.P. (a₁, a₂, ...., a_n) is having odd number of terms, then
         G.M. = (a₁ × a_n)^1/2 = (a₂ × a_{n - 1})^1/2 = ..... = a_k
    where, a_k = mid-term of G.P., and,     k = 

    	n + 1
    	2

    
    
    
12. If an G.P. (a₁, a₂, ...., a_n) is having even number of terms, then
         G.M. = (a₁ × a_n)^1/2 = (a₂ × a_{n - 1})^1/2 = ..... = (a_k × a_{k + 1})^1/2
    where, k = 

    	n
    	2

    
    
    

---

• Harmonic Progression:
  
  
1. H.P. = a₁, a₂, a₃, ....., a_n
   
   where, 

   	1
   	a1

   , 

   	1
   	a2

   , 

   	1
   	a3

   , ......., 

   	1
   	an

    are in A.P.
   
   
2. Harmonic mean: H.P. of three terms: a, b, c is b(mid-term).
   Here, H.P. (b) = 

   	2ac
   	(a + c)

   
   
   

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