Question

Let a,b,c>0 and a,b,c are in A.P. a^2,b^2,c^2 are in G.P. then choose the correct one ? (a)a=b=c, (b)a^2+b^2=c^2 , (c) a^2+c^2=3 b^2, (d) none of these

Answer 1

`a=b=c`

Explanation:

The generic terms of an AP sequence can be represented by:

`sf({a, a+d, a+2d })`

We are told that `{a,b,c}`, and we note that if we take a higher term and subtract its previous term we get the common difference; thus

`c-b = b-a`
`:. 2b = a+c` ..... [A]

The generic terms of an GP sequence can be represented by:

`sf({a, ar, ar^2 })`

We are told that `{a^2, b^2, c^2}`, and we note that if we take a higher term and divide by its previous term we get the common ratio, thus:

`c^2/b^2 = b^2/a^2 => c/b = b/a \ \ \ \` (as `a,b,c gt 0`)
`:. b^2 = ac` ..... [B]

Substituting [A] into [B] we have:

`((a+c)/2)^2 = ac`
`:. a^2 + 2ac + c^2 = 4ac`
`:. a^2 - 2ac + c^2 = 0`
`:. (a-c)^2 = 0`
`:. a=c`

And if we substitute `a=c` into Eq [B], we have:

`b^2 = c^2 => b= c \ \ \ \` (as `a,b,c gt 0`)

Hence we have `a=c` and `b=c => a=b=c`