Question

If a, b, c are in AP find 1. 4a,4b,4c are in ap2. A+4,b+4,c+a are in AP

Answer 1

The conventional approach is the simplest.
As a, b, c are in AP, we can say that 2b = a + c.

Now, 
(b + c) + (a + b) 
= 2b + a + c
= a + c + a + c                          (we have just found out that 2b = a + c)
= 2(a + c)

=> (b + c), (c + a) and (a + b) are in AP