Question

The sum of three numbers in the A.P. is – 3, and the product of these three numbers is equal to 8. Then find these three numbers.

Answer 1

Hint: Let us assume the three terms of the A.P such that when we add these three terms then we are left with only one variable So, that we can find the value of that variable by putting the sum of three terms equal to zero.

Complete Step-by-Step solution:

Now, as we know that the common difference of an A.P is the difference of any two the consecutive terms of that A.P.
So, now we have two find three terms of an A.P which follow the given conditions.
Let the second term of the three terms be a.
And the common difference of the A.P will be d.
So, second term – first term = d
So, a – first term = d
First term of the A.P will be a - d.
And, third term – second term = d
Third term – a = d
Third term = a + d
So, the required three terms of the A.P will be a – d, a, a + d.
Now as we know that the sum of the three terms is equal to – 3.
So, (a – d) + (a) + (a + d) = – 3
3a = – 3
a = – 1
As it is given in the question that the product of these three terms is equal to 8.
So, (a – d)*(a)*(a + d) = 8
Now solving the above equation. We get,
\[
  \left( {{a^2} - ad} \right)*\left( {a + d} \right) = 8 \\
  {a^3} + {a^2}d - {a^2}d - a{d^2} = 8 \\
  {a^3} - a{d^2} = 8 \\
\]
Now we had to put the value of a = - 1 in the above equation. We get,
\[ - 1 + {d^2} = 8\]
Now adding 1 to both sides of the above equation we get,
\[
  {d^2} = 9 \\
  d = \pm 3 \\
\]
Now d can have two values 3 or – 3. But a = - 1.
So, now we had to find the three terms of the A.P
If d = 3
First term of the A.P will be = a – d = – 1 – 3 = – 4.
Second term of the A.P will be = a = – 1
Third term of the A.P will be = a + d = – 1 + 3 = 2.
If d = – 3
First term of the A.P will be = a – d = – 1 –(– 3) = 2.
Second term of the A.P will be = a = – 1.
Third term of the A.P will be = a + d = – 1 – 3 = – 4.
Hence, the three terms of the A.P such that their sum is – 3 and product is 8 is equal to (– 4, – 1, 2) or (2, –1, – 4).

Note: Whenever we come up with a type of problem then first, we have to assume the middle term is equal to any variable and then we assume first and third term such that they are less and more than middle term by a common difference d respectively. After adding three terms we get the value of the middle term(b) and then we multiply all three terms to get the value of d. After that putting the value of a and d we will get the three terms of the A.P.