📘 Class 10 Maths Chapter 17: Arithmetic Progressions – Notes & Blog Post

 

🔢 Introduction

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant.

📌 Example: 2, 5, 8, 11, 14...
Here, the common difference (d) = 3.

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🧠 Important Terms

• First Term (a): The first number in the AP.

• Common Difference (d): Difference between any two consecutive terms.

  $$d = a_2 - a_1 = a_3 - a_2 = ...$$

• nth Term (Tₙ): The general formula to find any term:

  $$T_n = a + (n - 1)d$$

• Number of Terms (n): The total number of terms in the AP.

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🔍 Sum of n Terms of an AP

To find the sum of the first n terms:

$$S_n = \frac{n}{2} [2a + (n - 1)d]$$

or

$$S_n = \frac{n}{2}(a + l)$$

where l is the last term.

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📝 Types of Problems

➤ 1. Finding the nth term

Q: Find the 15th term of the AP: 3, 6, 9, 12...
A:

$$a = 3,\ d = 3,\ n = 15 T_{15} = a + (n - 1)d = 3 + (14)×3 = 3 + 42 = \boxed{45}$$

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➤ 2. Finding number of terms when last term is given

Q: How many terms are there in the AP: 7, 10, 13... up to 97?
A:

$$a = 7,\ d = 3,\ l = 97 T_n = a + (n - 1)d = 97 7 + (n - 1)×3 = 97 (n - 1)×3 = 90 → n - 1 = 30 → n = \boxed{31}$$

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➤ 3. Sum of n terms

Q: Find the sum of first 20 terms of the AP: 5, 8, 11...
A:

$$a = 5,\ d = 3,\ n = 20 S_n = \frac{n}{2} [2a + (n - 1)d] = \frac{20}{2}[2×5 + 19×3] = 10[10 + 57] = 10×67 = \boxed{670}$$

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📌 Key Formulas

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✅ Quick Tips

• Common difference (d) can be positive, negative, or zero.

• If the nth term becomes negative in a context-based question, recheck values.

• Use appropriate formulas based on given values (a, d, n, l).

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