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Arithmetic progressions are lists where you add the same number each time: 7, 11, 15, 19 and so on. Class 10 questions ask two things about them, a particular term and the sum of several terms, and wrap them in stories about savings, rows of seats or instalments. This post gives both formulas with where they come from, works four examples, and lists the slips that cost marks.
What does the syllabus expect?
CBSE's Class X Mathematics syllabus for 2026-27 lists the "derivation of the nth term and sum of the first n terms of AP and their application in solving daily life problems". So you may be asked to derive a formula, not only use it. The NCERT chapter, reprinted for 2026-27, gives both derivations and has an optional exercise marked as not from the examination point of view.
Where do the two formulas come from?
In an AP with first term a and common difference d, the terms are a, a + d, a + 2d, and so on. The second term has one d, the third has two, so the nth term has n − 1 of them:
aₙ = a + (n − 1)d
For the sum, write Sₙ forwards and backwards and add. Each pair of terms adds to the same total, first term plus last term, and there are n pairs, so 2Sₙ = n(a + l):
Sₙ = n/2 (a + l) = n/2 [2a + (n − 1)d]
Use the first form when you know the last term l, and the second when you know d.
Worked examples
Example 1. Which term of 7, 11, 15, ... is 135?
a = 7, d = 4, aₙ = 135.
7 + (n − 1)4 = 135, so (n − 1)4 = 128 and n − 1 = 32.
n = 33. The 33rd term is 135.
Example 2. Find the sum of the first 20 terms of the same AP.
S₂₀ = 20/2 [2(7) + 19(4)] = 10 (14 + 76) = 10 × 90 = 900.
Example 3. A savings plan. Meera saves ₹50 in the first week and ₹10 more each week than the week before. After how many weeks will her total savings reach ₹2000?
a = 50, d = 10, Sₙ = 2000.
n/2 [100 + (n − 1)10] = 2000, so n/2 (90 + 10n) = 2000.
Simplify: 5n(9 + n) = 2000, so n² + 9n − 400 = 0.
Factorise: (n + 25)(n − 16) = 0, so n = 16 or n = −25.
Reject −25, because the number of weeks cannot be negative.
In week 16 she saves 50 + 15 × 10 = ₹200. Check: S₁₆ = 16/2 (50 + 200) = 8 × 250 = 2000.
Answer: 16 weeks.
Example 4. From a sum to a term. The sum of the first n terms of an AP is Sₙ = 3n² + 2n. Find the nth term.
aₙ = Sₙ − Sₙ₋₁.
Sₙ₋₁ = 3(n − 1)² + 2(n − 1) = 3n² − 6n + 3 + 2n − 2 = 3n² − 4n + 1.
aₙ = (3n² + 2n) − (3n² − 4n + 1) = 6n − 1.
Check: S₁ = 5, so a₁ = 5 = 6(1) − 1. S₂ = 16, so a₂ = 11 = 6(2) − 1.
Why study worked examples first?
When a topic is new, studying a full solution before you try one is efficient. In a series of experiments, Sweller and Cooper (1985) found that students who studied worked algebra examples took less time on them and then solved similar problems faster and with fewer errors than students who only solved problems. The gain was for problems with the same structure, so follow worked examples with practice of the same type before you mix.
Which mistakes lose marks?
Using n instead of n − 1. See the callout.
Wrong sign of d. In 21, 18, 15, ... d = −3. Write d = second term − first term every time.
Answering with a term when a sum was asked. "How much in week 16?" is a term. "How much in total?" is a sum. Underline which one the question wants.
Keeping impossible values. In Example 3 the scheme expects the line rejecting −25.
Squaring slips in Sₙ₋₁. Expand (n − 1)² fully as n² − 2n + 1 before multiplying.
Mixing up l and the nth term. If the question gives the last term, use Sₙ = n/2 (a + l) and save time.
If you can follow solutions but keep slipping on your own, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can compare your working with each step.
A one-week practice routine
Day 1: Derive both formulas from memory, then check against NCERT.
Day 2: Study two worked examples for nth term, then do five on your own.
Day 3: Same for sums: two examples, five problems.
Day 4: Three word problems with a quadratic inside, like Example 3. Reject the impossible root in words.
Day 5: Two Sₙ-to-aₙ questions, then a mixed set of six.
For APs at JEE level, read sequences and series for JEE. For how board answers are marked, see common Class 10 maths board mistakes.
Today, derive Sₙ by writing the sum forwards and backwards, without looking at the book.
