If first term of an A.P. is a, second term is b and last term is c, then show that sum of all terms is

Given first term = a

Second term = b

Last term = c

Common difference d = second term – first term = b – a

We will first find the number of terms

We use n^th term of an A.P. formula

t_n = a + (n – 1)d

where n = no. of terms

a = first term

d = common difference

t_n = n^th terms

Thus, on substituting all values we get,

⇒ c = a + (n – 1)(b – a)

⇒ c = a + (b – a)n + a – b

⇒ c = 2a – b + (b – a)n

⇒ (b – a)n = c + b – 2a

Using Sum of n terms of an A.P. formula

where n = no. of terms

S_n = sum of n terms

On substituting all the values we get,

Hence, proved