Three numbers are in AP with common difference equal to 1/2. If the sum of the numbers is 6, then their product is P.

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Understand the Problem

The question is asking to find the product of three numbers that are in arithmetic progression (AP) with a common difference of 1/2, given that the sum of these numbers is 6. The product is denoted as P.

Answer

$15$
Answer for screen readers

The value of $2P$ is $15$.

Steps to Solve

  1. Define the three numbers in AP

Let the three numbers in arithmetic progression be:

  • First number: $x - d$
  • Second number: $x$
  • Third number: $x + d$

Given that the common difference, $d$, is $\frac{1}{2}$, the numbers can be represented as:

  • First number: $x - \frac{1}{2}$
  • Second number: $x$
  • Third number: $x + \frac{1}{2}$
  1. Set up the equation for the sum

The sum of these numbers is given to be 6: $$ (x - \frac{1}{2}) + x + (x + \frac{1}{2}) = 6 $$

Combine like terms: $$ 3x = 6 $$

  1. Solve for x

Divide both sides by 3: $$ x = 2 $$

Now substitute $x$ back to find the three numbers:

  • First number: $2 - \frac{1}{2} = \frac{3}{2}$
  • Second number: $2$
  • Third number: $2 + \frac{1}{2} = \frac{5}{2}$
  1. Calculate the product, P

The product of these three numbers is: $$ P = \left(\frac{3}{2}\right) \times 2 \times \left(\frac{5}{2}\right) $$

Calculate step by step:

  • Calculate $\left(\frac{3}{2} \times 2\right) = 3$
  • Then calculate $3 \times \left(\frac{5}{2}\right) = \frac{15}{2}$
  1. Find the value of $2P$

Now, find $2P$: $$ 2P = 2 \times \frac{15}{2} = 15 $$

The value of $2P$ is $15$.

More Information

This problem demonstrates how to work through an arithmetic progression and calculate both the sum and product of its elements. The use of the common difference and setting up equations based on given conditions are useful techniques in algebra.

Tips

  • Confusing the order of the numbers in AP which can lead to incorrect calculations.
  • Failing to use the common difference properly in defining the sequence.
  • Mistaking the sum or the product; ensure careful handling of each operation.

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