Suppose `a,b,c` are in AP and `a^(2),b^(2),c^(2)` are in GP, If `agtbgtc` and `a+b+c=(3)/(2)`, than find the values of a and c.

Explanation

Since, a,b,c are in AP and sum of a,b,c is given.
Let `a=b-D,c=b+D " " [Dlt0][therefore agtbgtc]`
and given `a+b+c=(3)/(2)`
`implies b-D+b+b+D=(3)/(2)`
` therefore b=(1)/(2)`
Then, `a=(1)/(2)-D` and `c=(1)/(2)+D`
Also, given `a^(2),b^(2),c^(2)` are in GP, than `(b^(2))^(2)=a^(2)c^(2)`
`implies pm b^(2)=ac implies pm (1)/(4)=(1)/(4)-D^(2)`
`impliesD^(2)=(1)/(4) pm (1)/(4)=(1)/(2) " "[therefore D ne 0]`
`therefore D=pm (1)/(sqrt2) implies D=-(1)/(sqrt2)" "[therefore Dlt0]`
Hence, `a= (1)/(2)+(1)/(sqrt2) " and " c=(1)/(2)-(1)/(sqrt2)`