CH – 5 ARITHMETIC PROGRESSIONS

Master step-by-step patterns and unlock high scores in your Class 10 board exams with confidence.

Chapter Overview

In this chapter, you will learn:

• What an Arithmetic Progression is: A list of numbers where each term increases or decreases by a fixed value.
• First term: The very first number that starts the sequence.
• Common difference: The fixed number added or subtracted between consecutive terms.
• Finding any term of an AP: How to calculate any future term without writing out the whole list.
• Finding the position of a term: How to determine which position a specific number holds in an AP.
• Finding the sum of terms: How to quickly add together many terms of a sequence.
• Solving simple problems based on AP: How to apply these rules to everyday situations like money savings and daily patterns.

🎯 Board Exam Focus

Practice these essential skills carefully:

• Identifying whether a given list of numbers is an Arithmetic Progression.
• Finding the first term from a given list or problem.
• Finding the common difference accurately using consecutive terms.
• Finding the nth term using the general formula.
• Finding which term has a given value by solving for n.
• Finding the sum of the first n terms using both main and last-term formulas.
• Solving real-life word problems based on Arithmetic Progressions.

Most Important Concepts

What is an Arithmetic Progression?

An Arithmetic Progression (or AP) is a sequence of numbers in which each term is found by adding a fixed number to the term before it, except for the first term.

Because you always add or subtract the exact same number, the gap between consecutive terms never changes.

Simple Examples of APs

• 1, 2, 3, 4, 5, … (Here, you always add 1)
• 100, 70, 40, 10, … (Here, you always subtract 30)
• 5, 5, 5, 5, … (Here, you always add 0)

Examples That Are NOT APs

• 1, 2, 4, 8, 16, …
  • Why it is not an AP: The gaps are 1, 2, 4, 8. The gap changes every time instead of staying the same.
• 1, 3, 7, 12, …
  • Why it is not an AP: The difference between terms keeps changing (3 − 1 = 2, 7 − 3 = 4).

Important Parts of an AP

Let us look at the key parts that define any AP:

• First term (a or a₁): The starting number of the sequence.
• Common difference (d): The fixed difference between two consecutive terms.
• Number of terms (n): The total count of terms being considered.
• nth term (aₙ): The term at position n.
• Last term (l): The final term of a finite AP (also written as aₙ).

Example

Take the AP: 4, 10, 16, 22, 28

Let us identify each part:

• First term (a): 4
• Common difference (d): 10 − 4 = 6
• Number of terms (n): 5 (there are 5 numbers in total)
• 3rd term (a₃): 16
• Last term (l): 28

Finding the Common Difference

To find the common difference d, subtract any term from the term that comes immediately after it:

d = a₂ − a₁ or d = aₙ − aₙ₋₁

1. Increasing AP (Numbers go up)

Consider: 3, 7, 11, 15, …

• d = 7 − 3 = 4
• When numbers increase, d is positive.

2. Decreasing AP (Numbers go down)

Consider: 20, 15, 10, 5, …

• d = 15 − 20 = −5
• When numbers decrease, d is negative.

Remember: Always subtract the earlier term from the later term (Second term − First term). Never reverse the order!

Finding the nth Term of an AP

To find any term at position n without writing down all terms, use the general term formula:

aₙ = a + (n − 1)d

What each letter means:

• aₙ = the value of the term at position n
• a = the first term
• n = the term position (must be a positive whole number)
• d = the common difference

Example Step-by-Step

Find the 10th term of the AP: 2, 7, 12, …

1. Write down what is given:
   • First term (a) = 2
   • Common difference (d) = 7 − 2 = 5
   • Term number (n) = 10
2. Write the formula:
   • aₙ = a + (n − 1)d
3. Put the values into the formula:
   • a₁₀ = 2 + (10 − 1) × 5
   • a₁₀ = 2 + (9) × 5
   • a₁₀ = 2 + 45
   • a₁₀ = 47

The 10th term is 47.

Finding Which Term Has a Given Value

When a question asks “Which term is…?”, you are given the value aₙ and you need to find the position n.

Example Step-by-Step

Which term of the AP 21, 18, 15, … is −81?

1. Write down what is given:
   • First term (a) = 21
   • Common difference (d) = 18 − 21 = −3
   • Term value (aₙ) = −81
2. Use the nth term formula:
   • aₙ = a + (n − 1)d
   • −81 = 21 + (n − 1)(−3)
3. Solve for n step-by-step:
   • −81 − 21 = (n − 1)(−3)
   • −102 = (n − 1)(−3)
   • Divide both sides by −3:
   • (−102) ÷ (−3) = n − 1
   • 34 = n − 1
   • n = 34 + 1
   • n = 35

Therefore, the 35th term is −81.

Sum of the First n Terms

The sum of the first n terms means adding all the terms together from the 1st term up to the nth term. We denote this sum as Sₙ.

Formula 1 (Main Formula)

Use this when you know a, d, and n:

Sₙ = (n / 2) × [2a + (n − 1)d]

Formula 2 (Short Formula)

Use this when you know the first term a and the last term l (or aₙ):

Sₙ = (n / 2) × (a + l)

Example Step-by-Step

Find the sum of the first 20 terms of the AP: 1, 4, 7, 10, …

1. Identify the values:
   • a = 1
   • d = 4 − 1 = 3
   • n = 20
2. Apply the main sum formula:
   • S₂₀ = (20 / 2) × [2(1) + (20 − 1) × 3]
   • S₂₀ = 10 × [2 + 19 × 3]
   • S₂₀ = 10 × [2 + 57]
   • S₂₀ = 10 × 59
   • S₂₀ = 590

The sum of the first 20 terms is 590.

📌 Important Formulas & Results

• Common Difference: d = a₂ − a₁
• nth Term (General Term): aₙ = a + (n − 1)d
• Sum of First n Terms (Standard): Sₙ = (n / 2) × [2a + (n − 1)d]
• Sum of First n Terms (with Last Term): Sₙ = (n / 2) × (a + l)
• Finding nth term using sum: aₙ = Sₙ − Sₙ⋁₁
• Sum of first n positive integers: Sₙ = n(n + 1) / 2

🧠 How to Know Which Formula to Use?

1. When to use the nth-term formula: aₙ = a + (n − 1)d

• Use this when a question asks to find a specific term like “Find the 15th term”.
• Use this when you need to find the first term a or common difference d from given terms.

2. When to find the value of n

• Use this when the question asks “Which term is equal to 100?” or “How many terms are in the AP?”.
• You set aₙ = 100 and solve the linear equation to find n.

3. When to use the main sum formula: Sₙ = (n / 2) × [2a + (n − 1)d]

• Use this when you need the total sum and you know the common difference d.
• Use this when given the total sum Sₙ and asked to find how many terms were added together.

4. When to use the last-term sum formula: Sₙ = (n / 2) × (a + l)

• Use this when the first term a and final term l are given directly, and you do not know or do not need to calculate d.

🧠 Concepts Students Find Difficult

1. Finding the correct common difference

Students often subtract the bigger number from the smaller number regardless of order. Always do: Second term − First term.

2. Understanding negative common difference

When an AP decreases (e.g., 10, 7, 4…), d is negative. Forgetting the minus sign will cause calculation errors.

3. Understanding what n means

n represents the position number. It must always be a positive whole number (1, 2, 3, …). If your calculation gives a fraction or negative number for n, that value cannot be a term of the AP.

4. Confusing the nth term (aₙ) with the sum (Sₙ)

• aₙ is the value of one single term at position n.
• Sₙ is the total value when you add all terms from position 1 to position n together.

5. Solving word problems

Translating real-world statements into mathematical variables (a, d, n, Sₙ) requires practice. Look for starting values (a), fixed regular changes (d), and final totals (Sₙ).

⚠️ Common Mistakes Students Make

1. Subtracting terms in the wrong order while finding d

• Mistake: For the AP 10, 6, 2, writing d = 10 − 6 = 4.
• How to avoid: Always subtract the earlier term from the later term: d = 6 − 10 = −4.

2. Using the wrong sign for negative differences

• Mistake: Writing aₙ = 20 + (n − 1) − 3 instead of using multiplication.
• How to avoid: Put negative values inside brackets: aₙ = 20 + (n − 1)(−3).

3. Forgetting the “minus one” in the nth-term formula

• Mistake: Writing aₙ = a + nd.
• How to avoid: Always write aₙ = a + (n − 1)d.

4. Confusing aₙ with Sₙ

• Mistake: Substituting the total sum value into the place of aₙ.
• How to avoid: Ask yourself: “Is this the single term at that place or the total sum of all terms?”

5. Accepting fractional or negative values for n

• Mistake: Getting n = 20.5 and stating that it is a term position.
• How to avoid: Remember that term counts must be counting numbers. If n is not a positive integer, the given number is not part of the AP.

🔥 Priority Revision

⭐⭐⭐ HIGH-PRIORITY PRACTICE

• Finding the nth term using aₙ = a + (n − 1)d.
• Finding n when the value of a term is given.
• Finding the sum using Sₙ = (n / 2) × [2a + (n − 1)d].
• Word problems involving daily life applications (savings, salary increments, rows of objects).

⭐⭐ IMPORTANT PRACTICE

• Checking whether a given sequence or condition forms an AP.
• Finding missing terms in a sequence when two non-consecutive terms are given.
• Using Sₙ = (n / 2) × (a + l) when the last term is known.

⭐ ADDITIONAL REVISION

• Finding terms from the end of an AP.
• Questions where aₙ = Sₙ − Sₙ⋁₁ is used.

⚡ Arithmetic Progressions Quick Revision Sheet

• AP Definition: Sequence where aₖ₊₁ − aₖ is constant for all values of k.
• First term: a
• Common Difference: d = a₂ − a₁ (can be positive, negative, or zero).
• General Form: a, a + d, a + 2d, a + 3d, …
• nth Term Formula: aₙ = a + (n − 1)d
• Sum Formula 1: Sₙ = (n / 2) × [2a + (n − 1)d]
• Sum Formula 2: Sₙ = (n / 2) × (a + l)
• Key Reminder 1: Position n must always be a positive integer (1, 2, 3, …).
• Key Reminder 2: For decreasing APs, d must be taken as negative.

📝 Test Yourself

1. Check whether the list of numbers 2, 4, 8, 16, … forms an AP. Give a clear reason for your answer.
2. Find the first term and common difference for the AP: −5, −1, 3, 7, …
3. Find the 15th term of the AP: 3, 8, 13, 18, …
4. Which term of the AP 7, 13, 19, … is 205?
5. Find the sum of the first 16 terms of the AP: 10, 6, 2, −2, …
6. Find the sum of all two-digit positive integers that are divisible by 5.
7. The 3rd term of an AP is 12 and the 7th term is 24. Find its 1st term and common difference.
8. How many terms of the AP 9, 17, 25, … must be taken to give a total sum of 636?
9. A student saves 20 rupees in the first week of a month and increases weekly savings by 10 rupees each week. How much money will be saved in total by the end of 6 weeks?
10. If the first term of an AP is 5 and the last term is 45, and the sum of all terms is 400, find the total number of terms.

🎯 Before Your Exam

☐ I can identify an Arithmetic Progression. ☐ I can find the first term of a given AP. ☐ I can calculate the common difference correctly. ☐ I know how to find the nth term using the general formula. ☐ I can find which term has a given value by solving for n. ☐ I can find the sum of terms using the sum formulas. ☐ I know which formula to use for different question types. ☐ I can solve AP word problems step-by-step.

You have all the tools needed to master this chapter—stay focused and keep practicing!