Question

if a, b, c are in A.P, then show that 1) b+c, c+a, a+b are also in A.P.

Answer 1

a , b , c are in A.P.



We know, 2 × middle term = sum of remaining two.

So,
➡ 2 × b = a + c
➡ 2b = a + c -------: ( 1 )




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

2 × ( c + a ) = b + a + b + c is true

Or, 2 × ( c + a ) = c + a + b + b is true

Or, 2 × ( c + a ) = c + a + 2b is true


$\bold{ \underline{ Putting \: value \: from \: ( i ) , }}$

Or, 2 × ( 2b ) = 2b + 2b is true

Or, 4b = 4b is true

0 = 0




Hence, both are equal. It means that b + c , c + a , a + b are in A.P.

Answer 2

❤❤Here is your answer ✌ ✌

$\huge\underline {\red {\bold {Answer}}}$

a,b,c are in ratio
We know, 2 × middle term = sum of remaining two. 

So, 
➡ 2 × b = a + c
➡ 2b = a + c -------: ( 1 ) 

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

2 × ( c + a ) = b + a + b + c is true

Or, 2 × ( c + a ) = c + a + b + b is true

Or, 2 × ( c + a ) = c + a + 2b is true 

Putting the value from(1)

Or, 2 × ( 2b ) = 2b + 2b is true 

Or, 4b = 4b is true 

0 = 0

Hence, both are equal. It means that b + c , c + a , a + b are in A.P