If the 3rd term of an arithmetic progression is 15 and the 7th term is 27, what

If the 3rd term of an arithmetic progression is 15 and the 7th term is 27, what is the common difference?

If the 3rd term of an arithmetic progression is 15 and the 7th term is 27, 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: Identify the formula for the nth term of an AP, which is given by: nth term = first term (a) + (n-1) * common difference (d).
Step 3: For the 3rd term, use n = 3. The equation becomes: a + 2d = 15.
Step 4: For the 7th term, use n = 7. The equation becomes: a + 6d = 27.
Step 5: Now you have two equations: a + 2d = 15 (Equation 1) and a + 6d = 27 (Equation 2).
Step 6: To find the common difference (d), first solve for 'a' in terms of 'd' from Equation 1: a = 15 - 2d.
Step 7: Substitute 'a' from Step 6 into Equation 2: (15 - 2d) + 6d = 27.
Step 8: Simplify the equation: 15 + 4d = 27.
Step 9: Solve for d: 4d = 27 - 15, which simplifies to 4d = 12.
Step 10: Divide both sides by 4 to find d: d = 12 / 4, which gives d = 3.

Arithmetic Progression – An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant.
Equations of Terms – The nth term of an AP can be expressed as a + (n-1)d, where a is the first term and d is the common difference.
Solving Linear Equations – The problem involves setting up and solving a system of linear equations to find the common difference.