Arithmetic Progressions

An arithmetic progression is a sequence where each new term obtained

by adding a common difference to the preceding term. If the first term

of the sequence is a then the arithmetic progression is

a, a + d, a + 2d, a + 3d, . . .n^th

ex- 4, 6, 8, 10..... n^th

where ,the n-th term is a + (n − 1)d. 

i.e

a_n= a+(n-1)d

a is first term

n is the term (not term value )

d is common value

a_n is the term value

 

                   

Let a be the first term and d be the common difference of an A. P. Let l denote the last term, i.e.,

the n-th term of the A. P. 

Then, l = a_n = a + (n – 1)

S_n denote the sum of the first n terms of the A. P. Then

S_n = a + (a + d) + (a + 2d) + ... + (l – 2d) + (l – d) + l............. (i)

Reversing the order of terms in the R. H. S. of the above equation, we have

S_n = l + (l – d) + (l –2d) + ... + (a + 2d) + (a + d) + ............... (ii)

Adding (i) and (ii) vertically, we get

2S_n = (a + l) + (a + l) + (a + l) + ... containing n terms = n (a + l)

Sn = (n/2)(a+l)

or

Sn = (n/2)[2a+(n–1)d]