The sum of the first three numbers in an AP is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers .?

Question

Answer

the terms are either 2,6,10 or 15,6,-3.

Step-by-step explanation:

It is given that the sum of first three numbers in an AP is 18. Product of the first and the third term is 5 times the common difference

Let first three numbers in the AP are a-d, a, a+d.

Sum of these three terms is 18.

Divide both sides by 3

The value of a is 6.

The product of the first and the third term is 5 times the common difference.

If the common difference is 4, then

Therefore the first three terms are 2, 6 and 10.

If the common difference is -9, then

Therefore the first three terms are 15, 6 and -3.

Answer

Answer

Given Information:
- The sum of the first three numbers in an arithmetic progression (AP) is 18.
- The product of the first and the third term is 5 times the common difference.

Let's solve the problem step by step:

Step 1: Understanding Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Step 2: Setting up the problem
Let's assume the first term of the AP as 'a', the second term as 'a + d', and the third term as 'a + 2d'. Here, 'a' represents the first term, and 'd' represents the common difference.

Step 3: Sum of the first three terms
The sum of the first three terms of an AP can be calculated using the formula:
Sum = (n/2)(2a + (n-1)d),
where 'n' is the number of terms.

In this case, the sum of the first three terms is given as 18.
So, we can set up the equation as:
18 = (3/2)(2a + 2d)
Simplifying the equation, we get:
18 = 3a + 3d
Dividing by 3, we have:
6 = a + d --- (Equation 1)

Step 4: Product of the first and third term
The product of the first and the third term can be expressed as:
Product = (a)(a + 2d)

According to the given information, the product is 5 times the common difference:
(a)(a + 2d) = 5d

Step 5: Solving the equations
Now, we have two equations:
Equation 1: 6 = a + d
Equation 2: (a)(a + 2d) = 5d

We can solve these equations simultaneously to find the values of 'a' and 'd'.

Step 6: Solution
Using Equation 1, we can express 'a' in terms of 'd':
a = 6 - d

Substituting this value in Equation 2, we get:
(6 - d)(6 - d + 2d) = 5d
Simplifying the equation:
(6 - d)(6 + d) = 5d
36 - d^2 = 5d
Rearranging the equation:
d^2 + 5d - 36 = 0
Factorizing the quadratic equation:
(d + 9)(d - 4) = 0
So, d = -9 or d = 4

If we substitute d = -9 in Equation 1, we get a negative value for 'a', which is not possible as it represents the first term of the AP. Therefore, we disregard d = -9 and consider d = 4.

Substituting d = 4 in Equation 1, we get:
6 = a + 4
a = 2

Thus, the three numbers in the arithmetic progression are:
First term (a) = 2
Second term (a + d) =

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