If the 3rd term of an arithmetic progression is 12 and the 7th term is 24, what

If the 3rd term of an arithmetic progression is 12 and the 7th term is 24, what is the common difference?

If the 3rd term of an arithmetic progression is 12 and the 7th term is 24, what is the common difference?

Step 1: Understand that in an arithmetic progression (AP), each term is found by adding a common difference (d) to the previous term.
Step 2: Let the first term of the AP be 'a'. The 3rd term can be expressed as 'a + 2d'.
Step 3: According to the question, the 3rd term is 12. So, we can write the equation: a + 2d = 12.
Step 4: The 7th term can be expressed as 'a + 6d'.
Step 5: According to the question, the 7th term is 24. So, we can write the equation: a + 6d = 24.
Step 6: Now we have two equations: a + 2d = 12 and a + 6d = 24.
Step 7: To find the common difference (d), we can subtract the first equation from the second: (a + 6d) - (a + 2d) = 24 - 12.
Step 8: Simplifying this gives us 4d = 12.
Step 9: Now, divide both sides by 4 to find d: d = 12 / 4.
Step 10: Therefore, d = 3.

Arithmetic Progression – Understanding the properties of arithmetic sequences, including how to find terms based on the first term and common difference.
Algebraic Manipulation – Ability to set up and solve equations based on given conditions.