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If you save ₹100 in the first month and increase your savings by ₹50 every month, how much money will you have accumulated in 10 years? Arithmetic Pro...
If you save ₹100 in the first month and increase your savings by ₹50 every month, how much money will you have accumulated in 10 years? Arithmetic Progressions predict linear sequences and sums.
Why This Chapter Matters
In Class 10 Mathematics, "Arithmetic Progressions" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Before You Begin (Prerequisites)
- Number patterns from Class 7.
- Linear equations.
- Series and sequences.
What You Will Learn (Core Objectives)
- Define an Arithmetic Progression (AP), first term ($a$), and common difference ($d$).
- Derive and apply the $n$-th term formula: $a_n = a + (n - 1)d$.
- Derive and apply the Sum of first $n$ terms formula: $S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}[a + l]$.
- Determine whether a given sequence forms an AP.
- Solve real-world financial installment and ladder problems using AP.
Chapter Roadmap & Progression
1
1. Definition of an AP
2
2. The $n$-th Term Formula
3
3. Sum of First $n$ Terms
Complete Concept Guide (100% Curriculum Coverage)
1. Definition of an AP
An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number $d$ (common difference) to the preceding term: $a, a+d, a+2d, a+3d\dots$ Common difference $d = a_k - a_{k-1}$ can be positive, negative, or zero.
2. The $n$-th Term Formula
The general $n$-th term is: $$\mathbf{a_n = a + (n - 1)d}$$ If an AP has $m$ terms, $a_m$ represents the last term $l$.
3. Sum of First $n$ Terms
The sum $S_n$ of the first $n$ terms of an AP is: $$\mathbf{S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}[a + l]}$$ Discovered intuitively by Carl Friedrich Gauss when asked to sum numbers from 1 to 100 as a schoolboy!
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
Find the 20th term of the AP: $4, 9, 14, 19\dots$
Reveal Answer & Explanation
Answer: $a = 4, d = 9 - 4 = 5$. $a_{20} = 4 + (20 - 1)(5) = 4 + 95 = 99$.
a_n = a + (n-1)d.
2
Which term of the AP $3, 8, 13, 18\dots$ is 78?
Reveal Answer & Explanation
Answer: $a_n = 78 \implies 3 + (n - 1)5 = 78 \implies 5(n - 1) = 75 \implies n - 1 = 15 \implies n = 16$. The 16th term.
n = 16.
3
Find the sum of the first 22 terms of the AP $8, 3, -2\dots$
Reveal Answer & Explanation
Answer: $a = 8, d = -5, n = 22$. $S_{22} = \frac{22}{2}[2(8) + (21)(-5)] = 11[16 - 105] = 11(-89) = -979$.
S_22 = -979.
4
If the sum of the first $n$ terms of an AP is given by $S_n = 3n^2 + 5n$, find its common difference.
Reveal Answer & Explanation
Answer: $S_1 = a_1 = 3(1) + 5(1) = 8$. $S_2 = a_1 + a_2 = 3(4) + 5(2) = 22 \implies a_2 = 22 - 8 = 14$. Common difference $d = a_2 - a_1 = 14 - 8 = 6$.
d = 6.
5
Find the sum of all two-digit odd positive integers.
Reveal Answer & Explanation
Answer: Sequence: $11, 13, 15\dots 99$. $a = 11, d = 2, l = 99$. $99 = 11 + (n-1)2 \implies 88 = 2(n-1) \implies n = 45$. $S_{45} = \frac{45}{2}(11 + 99) = \frac{45}{2}(110) = 45 \times 55 = 2,475$.
Sum = 2475.
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