Series and Arithmetic Progression (A.P)

A sequence or series is called an A.P if each term is obtained by adding a constant number to the previous term.  The constant is called the common difference

The nth term of an A.P

If the first term of an A.P is a and the common difference (d) Then the first four terms are U₁=a,  U₂=a+d, U₃=a+2d and U₄ = a+3d in general Un=a+ (n-1) d

Example

1. Find the (a) 10^th (b) 15^th (c) 100^th (d) nth term of A.P  5, 10, 15, 20……..

Solution

5, 10, 15, 20…………                                    (c)       U₁₀₀ = a+99d

= 5 + 495

= 500

• U₁₀ = a +9d (d)       U_n = a + (n-1) d

= 5 + 45                                                         = 5 + (n-1)5

= 50                                                                = 5 + 5n – 5

U_n = 5n

• U₁₅ = a+14d

= 5 + 70

= 75

2. Determine the number of the term which is 83 in the A.P 3, 8, 13, 18,…………

Solution

a=3, d=5, n=?   Un = 83

Un = a+(n-1)d

83 = 3 + 5(n-1)

5n – 2 = 83

5n = 85

n=17

Exercise:

In an AP the difference between the 8^th and 4^th term is 20 and 8^th term is  times the 4^th term.  Find the (i) common difference             (ii) the first term of the sequence.