Split 207 into three parts in AP such that the product of the two smaller parts is 4623.
67, 69, 71
Step 1: Let the three parts in Arithmetic Progression (AP) be \((a - d), a, (a + d)\).
This means the middle term is \(a\), the first term is smaller by \(d\), and the last term is larger by \(d\).
Step 2: According to the question, the sum of the three parts is 207.
So, \((a - d) + a + (a + d) = 207\).
Simplify: \(3a = 207\).
Therefore, \(a = \dfrac{207}{3} = 69\).
Step 3: Now the three numbers are \((69 - d), 69, (69 + d)\).
Step 4: The product of the two smaller parts is given as 4623.
The two smaller parts are \((69 - d)\) and 69.
So, \(69 \times (69 - d) = 4623\).
Step 5: Expand the equation:
\(69 \times 69 - 69d = 4623\).
\(4761 - 69d = 4623\).
Step 6: Bring terms together:
\(4761 - 4623 = 69d\).
\(138 = 69d\).
Step 7: Solve for \(d\):
\(d = \dfrac{138}{69} = 2\).
Step 8: Substitute \(d = 2\):
First part = \(69 - 2 = 67\).
Second part = \(69\).
Third part = \(69 + 2 = 71\).
Final Answer: The three parts are 67, 69, 71.