🔹 1. Introduction

A sequence is an ordered list of numbers following a specific rule, while a series is the sum of terms of a sequence.

👉 Example:

• Sequence: 2, 4, 6, 8, …
• Series: 2 + 4 + 6 + 8 + …

This chapter is extremely important for:

• Algebra
• Calculus (future topics)
• JEE Main & Advanced

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🔹 2. Sequence

✔️ Definition

A sequence is a function whose domain is natural numbers.

f: ℕ → ℝ

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✔️ Types of Sequences

1. Finite Sequence

Has limited terms
Example: 1, 2, 3, 4

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2. Infinite Sequence

Continues forever
Example: 1, 2, 3, …

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✔️ General Term (nth Term)

Denoted by aₙ

Example:
aₙ = 2n → 2, 4, 6, 8 …

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🔹 3. Series

✔️ Definition

Sum of sequence terms:

Sₙ = a₁ + a₂ + a₃ + … + aₙ

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🔹 4. Arithmetic Progression (AP)

✔️ Definition

A sequence where the difference between consecutive terms is constant.

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✔️ Standard Form

a, a + d, a + 2d, a + 3d, …

Where:

• a = first term
• d = common difference

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✔️ nth Term of AP

$a_n = a + (n-1)d$an​=a+(n−1)d

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✔️ Sum of First n Terms

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

Alternative form:

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

(where l = last term)

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✔️ Important Results

• nth term from end:
  = l − (n−1)d
• Sum of n terms from end = same as beginning

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✔️ Example

Find 10th term:

a = 2, d = 3

a₁₀ = 2 + 9×3 = 29

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🔹 5. Arithmetic Mean (AM)

If a, b, c are in AP:

b = (a + c)/2

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✔️ Inserting Arithmetic Means

Between a and b, if n AMs are inserted:

Common difference:

d = (b − a)/(n+1)

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🔹 6. Geometric Progression (GP)

✔️ Definition

A sequence where ratio of consecutive terms is constant.

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✔️ Standard Form

a, ar, ar², ar³, …

Where:

• a = first term
• r = common ratio

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✔️ nth Term of GP

$a_n = ar^{n-1}$an​=arn−1

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✔️ Sum of n Terms (r ≠ 1)

$S_n = a \frac{r^n – 1}{r – 1}$Sn​=ar−1rn−1​

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✔️ Special Case (r < 1)

$S_n = a \frac{1 – r^n}{1 – r}$Sn​=a1−r1−rn​

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✔️ Infinite GP (|r| < 1)

$S_\infty = \frac{a}{1 – r}$S∞​=1−ra​

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✔️ Example

a = 2, r = 3

a₄ = 2 × 3³ = 54

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🔹 7. Geometric Mean (GM)

If a, b, c are in GP:

b² = ac

So:

b = √(ac)

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🔹 8. Relationship Between AM and GM

For positive numbers:

$AM \ge GM$AM≥GM

Equality holds when numbers are equal.

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🔹 9. Sum to n Terms of Special Series

✔️ 1. Sum of First n Natural Numbers

$\frac{n(n+1)}{2}$2n(n+1)​

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✔️ 2. Sum of Squares

$\frac{n(n+1)(2n+1)}{6}$6n(n+1)(2n+1)​

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✔️ 3. Sum of Cubes

$\left(\frac{n(n+1)}{2}\right)^2$(2n(n+1)​)2

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🔹 10. Important Types of Series

✔️ 1. Arithmetic Series

Sum of AP terms

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✔️ 2. Geometric Series

Sum of GP terms

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✔️ 3. Harmonic Progression (HP)

If reciprocals are in AP:

Example:
1/a, 1/b, 1/c

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🔹 11. Finding Missing Terms

✔️ In AP

Use:

• aₙ formula
• Sₙ formula

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✔️ In GP

Use:

• aₙ = arⁿ⁻¹

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🔹 12. Applications (Important for Exams)

✔️ Finding sums quickly
✔️ Pattern recognition
✔️ Financial calculations
✔️ Physics formulas

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🔹 13. JEE & CBSE Important Points

✔️ Learn all formulas thoroughly
✔️ Practice mixed problems (AP + GP)
✔️ Infinite GP is very important
✔️ AM ≥ GM frequently asked
✔️ Word problems are common

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🔹 14. Common Mistakes

❌ Using wrong formula
❌ Confusing AP and GP
❌ Ignoring r < 1 condition
❌ Calculation errors in powers

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🔹 15. Practice Questions

1. Find nth term of AP
2. Find sum of 20 terms
3. Insert arithmetic means
4. Find GP sum
5. Evaluate infinite GP
6. Prove AM ≥ GM

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🔹 16. Quick Revision Sheet

• AP: constant difference
• GP: constant ratio
• aₙ(AP) = a + (n−1)d
• aₙ(GP) = arⁿ⁻¹
• Sₙ(AP) = n/2[2a + (n−1)d]
• Sₙ(GP) = a(rⁿ−1)/(r−1)
• S∞ = a/(1−r), |r| < 1

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🔹 17. Conclusion

Sequences and Series is a high-scoring and concept-based chapter. Mastering formulas + practice can easily fetch full marks in CBSE and strong weightage in JEE.