Arithmetic Progression (AP) in CBSE Class 10 Mathematics is one of the highest-scoring chapters, yet word problems in NCERT Exercises 5.2 and 5.3 frequently trip students up during board examinations. The most frequent error is confusing an event term — the value on a specific day or step (a_n) — with the cumulative total (S_n). In this comprehensive guide, we deconstruct the formulas, solve classic exam word problems step-by-step, and provide the exact examiner-approved presentation required for 100% marks.
Direct Answer Injection: What is an Arithmetic Progression?
An Arithmetic Progression (AP) is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference (d) to the preceding term. The general form is a, a + d, a + 2d, ..., a + (n - 1)d. If d > 0, the AP is increasing; if d < 0, it is decreasing; and if d = 0, all terms remain identical.
1. Core Formulas & When to Use Them
CBSE Class 10 AP is built upon two master equations. Understanding their distinct real-world physical meanings is vital:
- The n-th Term Formula:
a_n = a + (n - 1)d
Use when: The question asks for the quantity produced, saved, or positioned in a single specific period (e.g., "production in the 10th year" or "height of the 15th step"). - The Sum of First n Terms (S_n):
S_n = (n/2) * [2a + (n - 1)d] = (n/2) * [a + l](where l = a_n is the last term)
Use when: The question asks for an accumulated total over multiple periods (e.g., "total production during the first 10 years" or "total money saved over 20 months").
2. Cumulative (S_n) vs Event (a_n) Semantic Identifier Matrix
Use this decision table whenever tackling NCERT Exercise 5.2 and 5.3 word problems to immediately choose the right formula:
| Problem Keyword / Phrasing | Target Variable | Formula to Deploy | Common Real-World Example |
|---|---|---|---|
| "In the k-th year / on day k" | Single Event (a_n) | a_n = a + (n - 1)d |
Salary in the 7th year of employment |
| "Total amount / in all / altogether" | Cumulative Sum (S_n) | S_n = (n/2) * [2a + (n - 1)d] |
Total savings accumulated across 12 months |
| "Find the number of rungs / terms" | Count (n) | n = ((l - a) / d) + 1 |
Number of wooden logs stacked in decreasing rows |
| "First negative term" | Inequality (a_n < 0) | a + (n - 1)d < 0 |
Finding when temperature drops below freezing point |
3. High-Yield Board Exam Word Problem Walkthroughs
Case Study 1: The TV Factory Production Problem (NCERT Ex 5.2, Q16/17 Type)
Problem: A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find: (i) the production in the 1st year, (ii) the production in the 10th year, and (iii) the total production in the first 7 years.
Step-by-Step Solution:
- Let the first year production be
aand uniform annual increase bed. - Given
a_3 = 600 &implies; a + 2d = 600— (Equation 1) - Given
a_7 = 700 &implies; a + 6d = 700— (Equation 2) - Subtracting Eq 1 from Eq 2:
4d = 100 &implies; d = 25. - Substitute
d = 25into Eq 1:a + 2(25) = 600 &implies; a = 550sets (Production in 1st year). - (ii) Production in 10th year (a_10):
a_10 = a + 9d = 550 + 9(25) = 550 + 225 = 775sets. - (iii) Total production in first 7 years (S_7):
S_7 = (7/2) * [a + a_7] = (7/2) * [550 + 700] = (7/2) * 1250 = 7 * 625 = 4375sets.
Case Study 2: The Spiral Perimeter Problem (NCERT Ex 5.3, Q17)
A spiral is made up of successive semicircles with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π = 22/7)
Examiner Tip: Perimeter of a semicircle is π * r. The lengths are: l_1 = π(0.5), l_2 = π(1.0), l_3 = π(1.5), ..., l_13 = π(6.5).
This forms an AP with a = 0.5π, d = 0.5π, and n = 13.
S_13 = (13/2) * [2(0.5π) + (13 - 1)(0.5π)] = (13/2) * [π + 6π] = (13/2) * 7π
Substitute π = 22/7: S_13 = (13/2) * 7 * (22/7) = 13 * 11 = 143 cm.
4. Frequently Asked Questions (FAQ)
Can the number of terms n in an AP be fractional or negative?
No. The number of terms n represents a counting index and must strictly belong to the set of positive natural numbers (n ∈ ℕ). If a quadratic equation for n yields a negative or non-integer root, discard it with the examiner note: "Since n is a counting number, n must be a positive integer."
How do I find a_n when the sum formula S_n is given?
Use the fundamental recurrence relationship: a_n = S_n - S_{n-1} (for n ≥ 2), and a_1 = S_1. For instance, if S_n = 3n² + 5n, calculate S_{n-1} = 3(n-1)² + 5(n-1), expand and subtract to find a_n = 6n + 2.
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