Question

If a,b,c are in A.P., prove that b+c,c+a,a+b are also in A.P​

Answer 1

Step-by-step explanation:

Solution:-

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

One might also think of simpler and smarter solutions. Such as

a, b, c are in AP. If we add b to all these terms, the resulting terms should also b in AP because the terms would still have the same common difference.

=> a + b, 2b, b + c are in AP

=> (a + b), (c + a), (b + c) are in AP. (because 2b = a + c)

hope this helps you ☺️