Write the condition under which three numbers a, b, c may be in Aand G

1.	Write the condition under which three numbers a, b, c may be in Aand G.P. both.
Answer» If we consider a set of three numbers a, b and c which are in AP, we know that a + c =2b. Now, if they are in GP, ac =b^2. Now we have to see if it is possible to have three numbers satisfy both the equations. If we express a as 2b - c and replace it in ac = b^2, it gives us (2b-c)*c = b^2 => 2bc – c^2 = b^2 => b^2 + c^2 – 2bc =0 => (b-c) ^2 =0 => b = c Also, as a = 2b-c = 2c-c = c we find that all the three number have to be equal. So a set of three numbers which are in AP as well as GP are a set of three equal numbers.

Write the condition under which three numbers a, b, c may be in Aand G.P. both.

Answer»

If we consider a set of three numbers a, b and c which are in AP, we know that a + c =2b. Now, if they are in GP, a*c =b^2.

Now we have to see if it is possible to have three numbers satisfy both the equations.

If we express a as 2b - c and replace it in a*c = b^2, it gives us (2b-c)*c = b^2

=> 2bc – c^2 = b^2

=> b^2 + c^2 – 2bc =0

=> (b-c) ^2 =0

=> b = c

Also, as a = 2b-c = 2c-c = c

we find that all the three number have to be equal.

So a set of three numbers which are in AP as well as GP are a set of three equal numbers.