53-03 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. 

E) Find t_n for an AP where t₅  = 19 & t₁₅  = -21.

Solution:

1) Let " a " be the first term & " d " be a common difference.
2) So, t_n = a + ( n - 1 ) d
3) According to the problem, 
            t₅   = a +  ( 4 )  d =  19
            t₁₅ = a + (14 )  d = -21
                  (-)  (-)               (+)
  --------------------------------------------
                            - 10  d  =  40
                                  -  d  =  4
                                     d  =  - 4
4) We know that 
            t₅   =  a +  ( 4 ) d =  19
                    a + 4 * (- 4) =  19
                            a  - 16 =  19
                                  a   =  19 + 16
                                  a   =  35
5) So,  t_n = a + ( n - 1 ) d
            t_n = 35 + ( n - 1 ) * (- 4)
            t_n = 35 - 4 n + 4
            t_n = 39 - 4 n
6) Answer: Here the n^th term of an AP is  t_n =  39 - 4 n

F) Find, how many 3-digit natural numbers are divisible by 3.

Solution:

1) We know that the lowest 3-digit number is 100 & the largest one is 999.
2) So the numbers divisible by 3 between 100 & 999 are
     102, 105, 108, ----, 996, 999.
3) Here a = 102, d = 3 and last term = 999.
4) So, from  t_n = a + ( n - 1 ) d, we have,
                  999 = 102 + ( n - 1 ) * 3
     3 *  ( n - 1 )  =  999 - 102 
     3 *  ( n - 1 )  =  897 
            ( n - 1 )  =  897 / 3
            ( n - 1 )  =  299
                    n    =  299 + 1
                    n    =  300
5) Answer: Here there are 300 three-digit numbers that are divisible by 3.

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

Click here for the next basics.

Anil Satpute