ARITHMETIC PROGRESSION

Arithmetic progression (AP) or arithmetic sequence of numbers in which each term after the first is obtained by adding a constant, d to preceding term. The constant d is called common difference.

An Arithmetic progression is given by a, (a + d), (a + d), (a + 2d), (a + 3d), ….. Where a = the first term, d = common difference

Example:

1, 3, 5, 7, …. Is an arithmetic progression (AP) with a 1 and d = 2

7, 13, 19, 25, …. Is an arithmetic progression (AP) with a = 7 and d = 6

nth term of an arithmetic progression tn = a + (n – 1)d where tn = nth term, a = the first term,  d = common difference

Example 1:

Find 10^th term in the series 1, 3, 5, 7, ….

a = 1

d = 3 – 1

   = 2

10^th term:

T₁₀ = a + (n – 1)d

T₁₀ = 1 + (10 – 1)2

T₁₀ = 1 + 18

T₁₀ = 19

Example 2:

Find 16^th term in the series 7, 13, 19, 25, ….

a = 7

d = 13 – 7

   = 6

16^th term:

T₁₆ = a + (n – 1)d

T₁₆ = 7 + (16 – 1)6

T₁₆ = 7 + 90

T₁₆ = 97

Sum of first n^th terms in an arithmetic progression

Example 3:

Find 4 + 7 + 10 + 13 + 16 + … up to 20 terms

a = 4

d = 7 – 4

   = 3

sums of the first 20 term

= n ÷ 2[2a + (n – 1)d]

= 20 ÷ 2 [2(4) + (20 – 1)3] 

= 10[8 + (19)3]

= 10(8 + 57)

= 650

Questions that you can try:

1. The first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3. Find a formula for the nth term and the value of the 50th term

2. The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term
  

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