THREE NUMBERS ARE IN AP AND THEIR SUM IS 24 IF THE FIRST NUMBER IS DECREASED BY ONE SECOND NUMBER IS DECREASED BY TWO THEN THE NEW NUMBERS ARE IN GP FIND THE NUMBERS
a + a+d + a+2d = 24
(a+d-2)/(a-1) = (a+2d)/(a+d-2)
To solve this problem, let's break it down step by step:
Step 1: Determine the common difference (d) of the arithmetic progression (AP).
Since the sum of the three numbers in the AP is given as 24, we can use the formula for the sum of an AP to find the common difference.
The sum of an AP is given by the formula: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
Plugging in the values, we get: 24 = (3/2)(2a + 2d)
Step 2: Set up equations for the modified numbers in the geometric progression (GP).
The first number decreased by one is (a - 1), and the second number decreased by two is (a + d - 2). Let's call them b1 and b2, respectively.
Step 3: Determine the common ratio (r) of the geometric progression.
Using the equation for the ratio of consecutive terms in a GP, we can set up the following equation: b2 / b1 = r.
Substituting the values, we get: (a + d - 2) / (a - 1) = r.
Step 4: Solve the system of equations to find the values of 'a', 'd', and 'r'.
We now have two equations:
1) 24 = (3/2)(2a + 2d)
2) (a + d - 2) / (a - 1) = r
Solving these equations simultaneously will give us the values of 'a', 'd', and 'r'.
Step 5: Substitute the values of 'a' and 'd' back into the original question to find the three numbers.
Once we have found the values of 'a' and 'd', we can substitute them back into the original AP to find the three numbers.
Following these steps, you should be able to find the numbers in the arithmetic progression (AP) and the geometric progression (GP).