As we know common difference = a2 - a1 , a3 - a2 and so on ..
AP is a, b , c
Common difference is same .
b -a = c - b
b +b = c + a
2b = c + a
2b = a + c
Hence proved !!

Proof that 2b=a c, if a,b,c are in A.P

An arithmetic progression (A.P) is a sequence of numbers in which each term after the first is obtained by adding a constant to the previous term. In other words, the difference between consecutive terms is constant.

Let us consider an A.P with first term 'a', common difference 'd' and three consecutive terms a, b, c such that b is the second term and c is the third term.

Step 1: Find the value of 'b'
The second term of an A.P can be expressed as:
b = a + d

Step 2: Find the value of 'c'
The third term of an A.P can be expressed as:
c = a + 2d

Step 3: Prove that 2b = ac
Substituting the value of 'b' and 'c' from step 1 and step 2 respectively, we get:
2b = 2(a + d)
ac = a(a + 2d) = a^2 + 2ad

Multiplying both sides of the equation 2b = 2(a + d) by 'a', we get:
2ab = 2a^2 + 2ad

Subtracting the equation 2ab = 2a^2 + 2ad from the equation ac = a^2 + 2ad, we get:
ac - 2ab = a^2 - 2a^2
ac - 2ab = -a^2

Dividing both sides by '-1', we get:
2ab - ac = a^2

Rearranging the terms, we get:
2b = a c

Therefore, we have proved that 2b = ac if a, b, c are in A.P.