An Arithmetic Progression is simply a list of numbers where you add the same amount each time — like 3, 7, 11, 15. Once you know two small formulas, this becomes one of the quickest chapters to score in. This page teaches it in plain language, with a plan and an original practice set (with answers you can reveal) at the end.
What This Chapter Covers
- What an arithmetic progression is
- The n-th term
- Sum of the first n terms
- Solving problems and word problems
Your Game Plan for This Chapter
- First — Understand the common difference and how to spot an AP.
- Next — Learn the n-th term and sum formulas — they solve almost everything.
- Last — Practise word problems, then attempt the practice set below.
Study Notes
1. What Is an Arithmetic Progression?
An AP is a sequence in which each term is obtained by adding a fixed number, called the common difference (d), to the previous term. Its general form is a, a + d, a + 2d, a + 3d, …, where a is the first term. You can always find d by subtracting any term from the one after it: d = a2 − a1.
A sequence is an AP only if the difference between consecutive terms is the same throughout. Check two or three gaps — if they match, it is an AP.
2. The n-th Term
To jump straight to any term without listing them all, use the n-th term formula. For example, in 3, 7, 11, 15, … (a = 3, d = 4), the 15th term is a15 = 3 + (15 − 1) × 4 = 3 + 56 = 59.
3. Sum of the First n Terms
To add up the first n terms, use the sum formula. These two formulas cover the whole chapter:
It is (n − 1)d, not n·d. Forgetting the “minus one” is the single most common error in this chapter — the 1st term uses (1 − 1)d = 0, so a1 = a.
4. Solving Problems and Word Problems
Most questions give you some information and ask for a term, the number of terms, or a sum. Write down a, d and what is asked, choose the right formula, and solve. Word problems — savings that grow by a fixed amount, seats increasing row by row — are just APs in disguise; find a and d, then apply the same formulas.
Almost every AP question reduces to “find a and d, then use a formula.” Get those two values first, and the rest is a one-line calculation.
Practice Worksheet
Try each question fully on your own first, then click Show Answer to check yourself.
Q1. Find the 15th term of the AP: 3, 7, 11, 15, …
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Q2. Which term of the AP 5, 8, 11, … is 68?
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Q3. Find the sum of the first 20 terms of the AP: 2, 5, 8, …
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Q4. The first term of an AP is 7 and its 13th term is 43. Find the common difference.
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Q5. A person saves ₹100 in the first month and increases the saving by ₹50 each month. How much is saved in the 10th month, and in total over 10 months?
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Once these feel easy, you have genuinely finished this chapter. Do not aim for perfect on the first try — aim for one more correct question than yesterday.