Question

if a b and c are in AP prove that b+c c+a and a+b are in AP

Answer 1

Answer:

hi,

Step-by-step explanation:

If a, b, c are in AP, prove that b + c, c + a, and a + b are also in AP.

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)

thanks

Answer 2

Step-by-step explanation:

Given a, b, c are in AP

Therefore, b-a=c-b

c+a-(b+c) =a+b-(c+a)

-->c+a-b-c=a+b-c-a

a-b=b-c

b-a=c-b

Therefore, a+b, b+c, c+a are in AP

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