CBSE Class 10 Maths Notes: Arithmetic Progressions

Definition & Motivation: What are Arithmetic Progressions?

Arithmetic Progressions (APs) are sequences of numbers where the difference between consecutive terms is constant. Imagine a pattern, a predictable growth or decline. That’s the essence of an AP! We study them because they model real-world scenarios, making predictions and solving problems easier. Think of regularly increasing salaries, steps in a staircase, or distances covered over time. Understanding APs provides a powerful tool for analyzing these situations.

Core Principles: The Building Blocks

An AP is defined by two key elements:

• First Term (a): The initial value of the sequence.
• Common Difference (d): The constant difference between consecutive terms (can be positive, negative, or zero). This is calculated as $d = a_{n+1} – a_n$, where $a_n$ represents the nth term.

Knowing ‘a’ and ‘d’ unlocks everything!

Formulas: Essential Equations

These formulas are your secret weapons!

• General Term (nth term): $a_n = a + (n-1)d$
  This formula helps you find any term in the sequence directly, without calculating all the preceding terms.
• Sum of the First n Terms (S_n): There are two important formulas for this:
  $S_n = \frac{n}{2} [2a + (n-1)d]$ or $S_n = \frac{n}{2} [a + a_n]$
  These formulas allow you to calculate the sum of the first ‘n’ terms efficiently. The second version is particularly useful when you already know the last term ($a_n$).

Derivation of $S_n$ formula: We can derive the formulas for $S_n$ by writing the sum forward and backward and adding them. This eliminates the need to add all ‘n’ terms individually.

Examples: Putting Knowledge into Action

Let’s see these concepts in action with practical examples:

• Salary Problems: A person gets a starting salary of ₹20,000 per month with an annual increment of ₹1000. Find their salary in the 10th year. (Apply the general term formula). Also, calculate their total earnings over the first 10 years (apply the sum formula).
• Distance Problems: A runner covers 5 km in the first hour and decreases his speed by 0.5 km each subsequent hour. How far will he run in the 8th hour? (General term) What total distance will he run in 8 hours? (Sum formula)
• Savings Problems: A person saves ₹100 in the first month, ₹150 in the second month, ₹200 in the third month, and so on. Find the total savings in 12 months. (Apply sum formula).
• Identifying APs: Given a sequence of numbers, determine if it’s an AP and then find its common difference and a specific term. For instance: 2, 5, 8, 11… (Determine if AP and then find the 20th term)

Tips for Problem Solving

• Identify ‘a’ and ‘d’: The first and most important step.
• Choose the Right Formula: Understand which formula is needed based on what’s given and what you need to find.
• Careful with Units: Make sure your units are consistent (e.g., all amounts are in the same currency, time in the same units).
• Practice, Practice, Practice: The more you solve problems, the easier it will become.