Solving Arithmetic Progression Problems: Step-by-Step Guide with Examples

Arithmetic progression (AP) problems involve sequences where each term is a fixed number of units away from the previous term, known as the common difference. These problems often require the application of formulas for the sum of terms and the general term rule.

Let's work through a detailed example to solve an arithmetic progression problem involving the sum of terms. We will break down the solution into manageable steps to make it clear and understandable.

Problem Statement

The sum of the first 9 terms of an AP is 72, and the sum of the next 4 terms is 71. Find the AP.

Step-by-Step Solution

Step 1: Sum of the First 9 Terms

The sum of the first ( n ) terms of an AP is given by:

[S_n frac{n}{2} times (2a (n - 1)d)]

For the first 9 terms:

[S_9 frac{9}{2} times (2a 8d) 72]

Divide both sides by 9:

[9 times frac{9}{2} times (2a 8d) 144]

[92a 72d 144]

Simplify:

[4a 8d 16 quad text{(1)}

Step 2: Sum of the Next 4 Terms

The next 4 terms are the 10th, 11th, 12th, and 13th terms. The sum of these terms is:

[S_{10 text{ to } 13} (a 9d) (a 10d) (a 11d) (a 12d) 4a 42d 71]

Express the equation:

[4a 42d 71 quad text{(2)}

Step 3: Solving the Equations Simultaneously

Substitute equation (1) into equation (2):

[48 - 4d 42d 71]

[48 38d 71]

[38d 23]

[d frac{23}{38} frac{3}{2}]

Step 4: Finding the First Term

Substitute ( d frac{3}{2} ) back into equation (1):

[4a 8 left(frac{3}{2}right) 16]

[4a 12 16]

[4a 4]

[a 1]

(Correction: the actual calculation shows ( a 2 ))

Step 5: Forming the AP

Given the first term ( a 2 ) and the common difference ( d frac{3}{2} ), the arithmetic sequence is:

[2, 3.5, 5, 6.5, 8, 9.5, 11, 12.5, 14, 15.5, 17, 18.5, 20]

Conclusion

The arithmetic progression (AP) that satisfies the given conditions is:

[2, 3.5, 5, 6.5, 8, 9.5, 11, 12.5, 14, 15.5, 17, 18.5, 20]

General Term and Summation Rules

The general term ( t_n ) of an arithmetic sequence can be expressed as:

[t_n a (n - 1)d]

The general summation rule ( S_n ) is: