52-02 Basics of Arithmetic & Geometric Progression

Click here for the previous basics.

In the Previous Blog, we had seen some important concepts of Arithmetic & Geometric Progression. 

In this Blog, we Describe the Important Formulas of an AP & GP.

1) t_n =  a + (n - 1) d 

2) S_n = n * (a + l)/2

3) S_n = n * [ 2 a + (n - 1) d ) ] / 2

Now we will see some examples:

Problems related to t_n =  a + (n - 1) d:

A) Find the n^th term of an AP 3, 5, 7, 9 ...


Solution:
1)  Here a = 3, d = (5 - 3) = 2 so, d = 2.
2)  We know that 

      t_n =  a + (n -1) d
          =  3 + (n -1) (2)
          =  3 + (2 n - 2)
          =  1 + (2 n)
          =  (2 n) + 1
3) Answer: Here the n^th term of an AP is  t_n =  (2 n) + 1 

B) Find the first term of an AP in which d = 4 and its 100^th term is 403.

Solution:
1)  Here d = 4 and t₁₀₀ =  403.
2)  We know that 

      t_n =  a + (n -1) d
   403 =  a + (100 - 1)*(4)
   403 =  a + (99)*(4)
   403 =  a + (396)
       a =  403 - 396
       a =  7
3) Answer: Here the 1^st term is a = 7

C) If the n^th term of an AP is m and the m^th term of an AP is n, then find the value of d.

Solution:
1)  Let " a " be the 1^st term and " d " be a common difference.
2)  We know that 

      t_n =  a + (n -1) d
3)  So we,
      t_n  =  a +  (n -1) d = m      ----------- (1)
      t_m =  a + (m -1) d = n       ----------- (2)
               Subtract equation (2) from (1) we get,
        a +  (n -1) d = m
        a + (m -1) d =  n  
     (-)  (-)               (-)  
----------------------------------
(n - 1 - m + 1) * d = (m - n)
           (n - m) * d = (m - n)
                         d = (m - n)/(n-m)
                         d = - 1
4) Answer: Here the common difference is d = - 1

In the next part, we will see a few examples and some essential formulae.

Click here for the next basics.

Anil Satpute