Arithmetic Progressions
1.0Master Linear Sequences, Common Differences, and Series Summation in Minutes
Unlock the mathematical logic governing predictable numeric patterns. Learn how to identify constant incremental steps, master the formula to find any distant term in a sequence, and calculate the total sum of large series to ace your Class 10 board exams.
2.0Learning Outcomes
After completing this lesson, you will be able to:
- Identify whether a given numeric sequence qualifies as an Arithmetic Progression.
- Extract the first term (a) and calculate the common difference (d) of any AP.
- Compute any specific term (an) using the standard general term formula.
- Calculate the sum of the first n terms (Sn) of an AP using both standard and last-term variations.
- Model and solve word problems based on real-world salary increments, savings ladders, and staging intervals.
3.0Introduction to the Arithmetic Progressions
Welcome to a highly structured and pattern-driven chapter of Class 10 Algebra! In daily life, we frequently encounter regular patterns: the rungs of a ladder, savings accounts growing by a fixed monthly deposit, or uniform seating rows in an auditorium. In mathematics, a sequence that increases or decreases by a fixed constant at each step is called an Arithmetic Progression (AP).
In this lesson, we unpack the DNA of linear sequences. You will learn how to identify the foundational First Term (a) and the unchanging Common Difference (d). We will master the algebraic formulas used to pinpoint any distant term (nth term) instantly without writing out the whole list, and discover how to add thousands of consecutive terms together in seconds using systematic series summation formulas.
Patterns are everywhere—from monthly savings and salary increments to staircase designs and seating arrangements. One of the most common mathematical patterns is an Arithmetic Progression (AP), where each term increases or decreases by a fixed value. Understanding AP is essential for CBSE Class 10 students and forms the basis for higher mathematics and competitive exams.
This comprehensive guide covers everything you need to know about Arithmetic Progression, including definitions, formulas, solved examples, real-life applications, previous year questions, FAQs, and revision notes.
An Arithmetic Progression (AP) is a sequence in which the difference between two consecutive terms remains constant.
Examples:
- 3, 6, 9, 12, 15...
- 25, 20, 15, 10...
- 100, 100, 100...
Each number is called a term, while the fixed value added or subtracted is called the common difference.
4.0Key Terms in Arithmetic Progression
5.0Common Difference (d)
The common difference is obtained by subtracting one term from the next.
d = Second Term − First Term
Examples: Sequence: 4, 9, 14, 19...
Common Difference
d = 9 − 4 = 5
The common difference can be: Positive, Negative and Zero
Here are some AP examples with their first term and common difference.
- 7, 14, 21, 28, 35, . . . . is an AP with the first term 7 and common difference 7.
- 91, 81, 71, 61, 51, . . . . is an AP with the first term 91 and common difference -10.
- π, 2π, 3π, 4π, 5π,… is an AP with the first term π and common difference π.
6.0Derivation of the Sum Formula of an Arithmetic Progression (AP)
Suppose an Arithmetic Progression has the first term a, common difference d, and n terms. Let the sum of the first n terms be Sₙ.
Step 1: Write the AP in the forward order
Sₙ = a + (a + d) + (a + 2d) + ... + [a + (n − 1)d]
The last term is l, where
l = a + (n − 1)d
Hence,
Sₙ = a + (a + d) + (a + 2d) + ... + l
Step 2: Write the AP in the reverse order
Sₙ = l + (l − d) + (l − 2d) + ... + a
Step 3: Add the two equations
Sₙ = a + (a + d) + (a + 2d) + ... + l
Sₙ = l + (l − d) + (l − 2d) + ... + a
------------------------------------------------
2Sₙ = (a + l) + (a + l) + (a + l) + ... + (a + l)
Since there are n terms,
2Sₙ = n(a + l)
Therefore,
Sₙ = n(a + l)/2 ...(1)
Step 4: Derive the formula in terms of the common difference
Since
l = a + (n − 1)d
Substitute this into Equation (1):
Sₙ = n/2 [a + a + (n − 1)d]
Sₙ = n/2 [2a + (n − 1)d] ...(2)
Final Formulas
- When the last term is known: Sₙ = n(a + l)/2
- When the common difference is known: Sₙ = n/2 [2a + (n − 1)d]
These two formulas are used to calculate the sum of the first n terms of an Arithmetic Progression quickly and efficiently.
7.0Important Notes on Arithmetic Progression (AP)
- An Arithmetic Progression (AP) is a sequence in which each term is obtained by adding a fixed number, called the common difference, to the previous term.
- The first term is denoted by a.
- The common difference is denoted by d.
- The nth term is denoted by aₙ.
- The number of terms is represented by n.
- The general form of an AP is:
a, a + d, a + 2d, a + 3d, ... - The nth term of an AP is given by:
aₙ = a + (n − 1)d - The sum of the first n terms of an AP is:
Sₙ = n/2 [2a + (n − 1)d] - If the last term (l) is known, the sum can also be calculated using:
Sₙ = n(a + l)/2 - An AP can be increasing, decreasing, or constant, depending on the value of the common difference.
- The common difference (d) may be positive, negative, or zero.
- If d > 0, the AP is increasing.
- If d < 0, the AP is decreasing.
- If d = 0, all the terms in the AP are equal.
- The graph of an Arithmetic Progression is a straight line, where the slope represents the common difference.
- Example of a decreasing AP:
16, 8, 0, –8, –16, ...
Here,
d = 8 − 16 = 0 − 8 = –8 − 0 = –16 − (–8) = –8 - Arithmetic Progressions are widely used in finance, construction, computer programming, physics, and everyday calculations involving constant increments or decrements.
8.0Solved Examples
Question: Which term of the AP 3, 8, 13, 18,... is 78?
Solution: The given progression is 3,8,13,18,...
Here the first term is a = 3, and the common difference is, d = 8 - 3= 13 - 8 = ... = 5
Let us assume that the nth term is,
an = 78
Put all these values in the general term of an arithmetic progression:
an = a+(n - 1)d
78 = 3 +(n - 1)5
78 = 3 + 5n - 5
78 = 5n - 2
80 = 5n
16 = n
Answer: ∴ 78 is the 16th term.
9.0Difference Between Arithmetic Progression (AP) and Geometric Progression (GP)
10.0Important topics in Class 10 Maths: Arithmetic Progression
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12.0Supporting Study Materials
This study material, containing comprehensive CBSE Notes and NCERT Solutions for Chapter 11 of Class 10 Maths, follows the latest NCERT guidelines. Complete with colorful sector partition charts, step-by-step segment deduction tracks, and angular rotation lookup blocks, this guide ensures thorough preparation for your board examinations.
30-Second Quick Review: Arithmetic Progression
- An Arithmetic Progression (AP) has a constant common difference.
- Common Difference: d = Second Term − First Term
- nth Term Formula: an=a+(n−1)d
- Sum of First n Terms: Sn=2n[2a+(n−1)d]
- Alternate Sum Formula: Sn=2n(a+l)
- If d > 0, the AP is increasing.
- If d < 0, the AP is decreasing.
- Arithmetic Progressions are widely used in finance, construction, scheduling, and pattern recognition.
13.0Previous Year Questions (PYQs) on Arithmetic Progression
Read the passage and answer the questions.
The number of spectators attending a sports tournament increases by 250 each day. On the first day, 1,500 spectators attend the event.
(i) Identify the type of sequence formed.
Answer: Arithmetic Progression
(ii) Find the common difference.
Answer: 250
(iii) How many spectators attend on the 10th day?
Answer: a10=1500+9x250
=3750
Answer: 3,750 spectators
(iv) Find the total number of spectators during the first 10 days.
Answer: S10=210(1500+3750)
=5 x 5250
=26,250
14.0Recommended Next Topics