An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms stays constant. AP formulas help students calculate the position of a term, the total number of terms, and the sum of a series. 

CBSE board questions often combine direct formula-based problems with word problems drawn from real situations, such as savings, distances, and arrangements. Accuracy in selecting values for the first term and the common difference decides the final answer.

This blog explains all the Arithmetic Progression formulas from CBSE Class 10 Chapter 5 using clear definitions.

Important Arithmetic Progression Formulas for Class 10

An arithmetic progression (AP) is a sequence where each term increases or decreases by a constant value called the common difference (d). Understanding AP formulas helps students solve questions on the nth term, the total terms, and the sum of terms, which frequently appear in CBSE exams. Below is a clear guide to all key formulas, terms, and applications.

1. Sequence: An ordered list of numbers, e.g., 2, 4, 6, 8 …
2. Arithmetic Progression (AP): A sequence where each term is obtained by adding a fixed number to the previous term, except the first.
3. First Term (a): The first number in the sequence.
4. Common Difference (d): The fixed value between consecutive terms. Can be positive, negative, or zero.
5. Nth Term (an): The general formula to find the nth term:
   an​=a+(n−1)d
6. Nth Term from the Last Term (l):
   an​=l−(n−1)d
7. Sum of First n Terms (Sn): When the last term is unknown:
   Sn​=n/2​[2a+(n−1)d]
   When the last term (an) is known:
   Sn​=n/2(a+An​)
8. Sum of First n Positive Integers:
   Sn​=1+2+3+…+n=2n(n+1)​
9. Arithmetic Mean (AM): For three numbers a, b, c in AP:
   b=a+c/2

Practical Applications of AP Formulas

1. Patterns in Nature: Many patterns in nature follow an arithmetic sequence. For example, the arrangement of petals in flowers, the spirals on pine cones, or the distribution of leaves on a stem can be studied using AP.
2. Finance and Pricing: AP formulas help in calculating totals where charges increase consistently. For example, taxi fares often have a fixed starting fee plus a constant amount per kilometer. Using AP, students can calculate the total fare for any distance.
3. Construction and Decoration: Arranging objects in layers or rows, like bricks, tiles, or decorations, often follows an AP. The total number of objects can be found easily using the sum formula.
4. Science and Research: AP is used in research fields like physics, space science, and nuclear science to study sequences, patterns, or evenly increasing quantities.
5. Scheduling and Planning: AP can be applied in daily planning, such as distributing work tasks evenly over time or arranging seats in rows with increasing counts.

AP Formula Solved Examples

Example 1 – Book Arrangement

Problem:
A librarian arranges books on a shelf. The first row has 8 books, and each subsequent row has 3 more books than the previous one. If there are 6 rows, how many books are there in total?

Solution:
AP: 8, 11, 14, …

Sum of 6 terms: S6​=26​[2(8)+(6−1)(3)]=3[16+15]=3×31=93

Answer: 93 books

Example 2 – Water Bottles

Problem:
A school distributes water bottles to students such that the first student gets 1 bottle, the second gets 2 bottles, the third 3 bottles, and so on. How many bottles are needed for 15 students?

Solution:
AP: 1, 2, 3, … S15​=215​[2(1)+(15−1)(1)]=215​[2+14]=215​×16=120

Answer: 120 bottles

Tips to Memorize AP Formulas

1. Understand the logic first: Learn why formulas work, not just memorizing them.
2. Practice examples: Solving multiple questions helps you remember the formulas naturally.
3. Use formula sheets: Keep a reference sheet for quick revision before exams.
4. Self-tests: Take mini-tests to identify areas needing improvement.
5. Visual patterns: Draw sequences or diagrams to see the arithmetic progression in action.