Sequences and Series Explained: Arithmetic, Geometric, and Sums
PUC Mathematics | Sequences & Series | Daniel Classes
Why This Chapter Matters
Sequences are lists of numbers following a pattern. Series are sums of those sequences. Together, they model compound interest, population growth, loan amortization, and countless real-world phenomena.
Yet most students treat this chapter as “memorize formulas, apply them, get the answer.” The real skill is recognizing patterns and understanding why the formulas work.
Core insight: Every arithmetic or geometric sequence has a closed-form formula. Once you know the pattern, you can find any term without calculating all the previous ones.
Sequences: Definitions
An Arithmetic Sequence
Definition: A sequence where the difference between consecutive terms is constant.
Example: 2, 5, 8, 11, 14, … (common difference d = 3)
General term: aₙ = a₁ + (n-1)d
Example: For the sequence above, a₅ = 2 + (5-1)(3) = 2 + 12 = 14 ✓
A Geometric Sequence
Definition: A sequence where the ratio between consecutive terms is constant.
Example: 2, 6, 18, 54, 162, … (common ratio r = 3)
General term: aₙ = a₁ × r^(n-1)
Example: For the sequence above, a₅ = 2 × 3^(5-1) = 2 × 81 = 162 ✓
Series: Sums of Sequences
Sum of an Arithmetic Series
Definition: Sₙ = a₁ + a₂ + … + aₙ
Formula: Sₙ = n/2 × (a₁ + aₙ) or Sₙ = n/2 × (2a₁ + (n-1)d)
Example: Sum of first 10 terms of 2, 5, 8, …:
S₁₀ = 10/2 × (2×2 + (10-1)×3) = 5 × (4 + 27) = 5 × 31 = 155
Sum of a Geometric Series
Definition: Sₙ = a₁ + a₂ + … + aₙ
Formula: Sₙ = a₁ × (1 – r^n) / (1 – r) [if r ≠ 1]
Example: Sum of first 5 terms of 2, 6, 18, …:
S₅ = 2 × (1 – 3^5) / (1 – 3) = 2 × (1 – 243) / (-2) = 2 × (-242) / (-2) = 242
Infinite Geometric Series
Definition: If |r| < 1, the series converges.
Formula: S∞ = a₁ / (1 – r)
Example: Sum of 1 + 0.5 + 0.25 + 0.125 + … = 1 / (1 – 0.5) = 2
How PUC Exams Test Sequences & Series
Type 1: Finding Terms (2-3 marks)
Example: “Find the 10th term of 2, 5, 8, …”
Solution: aₙ = 2 + (10-1)(3) = 29
Type 2: Sum of Finite Series (3-5 marks)
Example: “Find the sum of the first 15 terms of 3, 6, 9, …”
Solution: S₁₅ = 15/2 × (2×3 + (15-1)×3) = 15/2 × (6 + 42) = 360
Type 3: Geometric Series (3-5 marks)
Example: “Find the sum of the first 6 terms of 1, 2, 4, 8, …”
Solution: S₆ = 1 × (1 – 2^6) / (1 – 2) = (1 – 64) / (-1) = 63
The Five Common Mistakes
Mistake 1: Confusing d (Difference) with r (Ratio)
Example: “For 2, 5, 8, 11, find the common difference.”
Wrong: “r = 5/2 = 2.5”
Right: “d = 5 – 2 = 3”
Fix: For arithmetic: subtract. For geometric: divide.
Mistake 2: Misusing the Index in the Formula
Example: “Find a₅ for 2, 5, 8, …”
Wrong: “a₅ = 2 + 5(3) = 17” [Using n = 5 directly in the multiplier]
Right: “a₅ = 2 + (5-1)(3) = 14”
Fix: The formula uses (n-1), not n.
Mistake 3: Forgetting the Denominator in Geometric Sum
Example: “Find S₅ for 1, 2, 4, 8, …”
Wrong: “S₅ = 1(1 – 2^5) = -31” [Forgot the (1-r) denominator]
Right: “S₅ = 1(1 – 2^5) / (1 – 2) = 31”
Fix: Always use the full formula: a₁(1 – r^n) / (1 – r).
Mistake 4: Using the Wrong Series Formula
Example: “Find S₆ for 2, 6, 18, …”
Wrong: Uses arithmetic formula (S = n/2 × …).
Right: Uses geometric formula (S = a₁(1 – r^n)/(1-r)).
Fix: Check the pattern. If the ratio is constant, use geometric. If the difference is constant, use arithmetic.
Mistake 5: Applying Infinite Series Formula When |r| ≥ 1
Example: “Find S∞ for 1, 2, 4, 8, …”
Wrong: “S∞ = 1 / (1 – 2) = -1” [r = 2, series diverges]
Right: “The series diverges (r = 2 > 1), so no sum exists.”
Fix: Infinite geometric series converges only if |r| < 1.
What to Do Next
To master arithmetic sequences: Find 10 different terms and sums for different arithmetic sequences.
To master geometric sequences: Find 10 different terms and sums, including some with |r| < 1 for convergence.
To master infinite series: Identify which geometric series converge, then find their sums.
The Problem Vault Alternative
The Problem Vault has 70+ Sequences & Series problems from actual PUC exams.