if a^{2}, b^{2},c^{2 }are in a A.P .then prove that the following are also in A.P (i) 1/b+c ,1/c+a ,1/a+b (ii) a/b+c , b/a+c ,c/b+a

(i) To prove 1/b+c , 1/a+c ,1/b+a are in A.P

Given that a^{2}, b^{2},c^{2 }are in a A.P

Therefore,

(i) To prove a/b+c , b/a+c ,c/b+a are in A.P

a/b+c , b/a+c and c/b+a will be in A.P if a = b = c.

If 3 numbers are in A.P and all the given numbers are multiplied by the same number then the resultant numbers will also be in A.P

For example:

Take 1, 2, 3 and multiply by 4, we get 4, 8 and 12 which are also in A.P or take 5, 7, 9 and multiply by 8, we get

40, 56 and 72 which are also in A.P.

However, you cannot multiply them by different numbers as this will never lead to A.P.

Therefore, a/b+c , b/a+c ,c/b+a will be in A.P if a = b = c.

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