Three numbers whose sum is 15 are in A.P. If 1,4,19 be added to them respectively the resulting number are in G.P find
To find the three numbers, let's assume the common difference in the arithmetic progression (A.P.) is 'd', and the first term is 'a'.
As per the given condition, the sum of three numbers in the A.P. is 15. So, we can form the equation:
a + (a+d) + (a+2d) = 15
Simplifying the equation, we get:
3a + 3d = 15
Dividing both sides by 3, we have:
a + d = 5
Now let's find the three numbers after adding 1, 4, and 19 respectively:
First number = a + 1
Second number = (a+d) + 4
Third number = (a+2d) + 19
According to the given condition, these resulting numbers are in geometric progression (G.P.). In a G.P., the ratio between consecutive terms is constant.
Let's assume the common ratio in the G.P. is 'r'. Therefore, we can form the following equations:
(a + 1)/(a + d) = (a + d + 4)/(a + 2d)
(a + d + 4)/(a + d) = (a + 2d)/(a + d + 19)
Simplifying these equations will give us the values of 'a', 'd', and 'r', and consequently, the three numbers.
Solving these equations can be a tedious process, so it would be helpful to use a symbolic algebra system or a graphing calculator to find the values of 'a', 'd', and 'r'.