Question

For which value of x the 2x, x + 8 and 3x + 1 will be in arithmetic progression ?

Hint:

For any three consecutive terms in an AP, the middle term is the arithmetic mean of the other two. This property, \( \text{Middle Term} = \frac{\text{First Term} + \text{Last Term}}{2} \), is the basis for the formula \(2t_2 = t_1 + t_3\) and is the fastest way to solve such problems.

Answer 1

Step 1: Understanding the Concept:
An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.
Step 2: Key Formula or Approach:
If three terms \(t_1, t_2, t_3\) are in an AP, then the common difference is the same.
\[ t_2 - t_1 = t_3 - t_2 \] This can be rearranged to a more convenient form:
\[ 2t_2 = t_1 + t_3 \] Step 3: Detailed Explanation:
We are given three terms in AP:
\( t_1 = 2x \)
\( t_2 = x + 8 \)
\( t_3 = 3x + 1 \)
Using the property \( 2t_2 = t_1 + t_3 \):
\[ 2(x + 8) = (2x) + (3x + 1) \] Now, we solve this linear equation for x.
Distribute the 2 on the left side:
\[ 2x + 16 = 2x + 3x + 1 \] Combine like terms on the right side:
\[ 2x + 16 = 5x + 1 \] Move the x terms to one side and the constant terms to the other:
\[ 16 - 1 = 5x - 2x \] \[ 15 = 3x \] Divide by 3:
\[ x = \frac{15}{3} \] \[ x = 5 \] Step 4: Final Answer:
The value of x for which the terms are in arithmetic progression is 5. We can check this: the terms are 2(5)=10, 5+8=13, 3(5)+1=16. The sequence 10, 13, 16 has a common difference of 3.