📊 Sequence Closed Forms — Finite GP/AP Sums, Telescoping

Essential for sum problems and sequence analysis in AMC contests.

🎯 Recognition Cues

  • “Find the sum” — Sum of sequence terms
  • “What is the $n$th term” — General term of sequence
  • Arithmetic sequence — Constant difference between terms
  • Geometric sequence — Constant ratio between terms
  • Telescoping — Most terms cancel out

📚 Template Solutions

Arithmetic Sequences

ConceptFormulaExample
$n$th term$a_n = a_1 + (n-1)d$$a_5 = 3 + 4 \cdot 2 = 11$
Sum of first $n$ terms$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$S_{10} = 5(6 + 9 \cdot 2) = 120$
Sum of first $n$ terms$S_n = \frac{n}{2}(a_1 + a_n)$$S_{10} = 5(3 + 21) = 120$

Geometric Sequences

ConceptFormulaExample
$n$th term$a_n = a_1 \cdot r^{n-1}$$a_4 = 2 \cdot 3^3 = 54$
Sum of first $n$ terms$S_n = a_1 \frac{1-r^n}{1-r}$$S_5 = 2 \cdot \frac{1-3^5}{1-3} = 242$
Infinite sum (if $r< 1$)

Telescoping Series

TypeMethodExample
Partial fractions$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$$\sum_{k=1}^n \frac{1}{k(k+1)} = 1 - \frac{1}{n+1}$
Difference of squares$n^2 - (n-1)^2 = 2n - 1$$\sum_{k=1}^n (2k-1) = n^2$
Trigonometric$\sin(A+B) - \sin(A-B) = 2\cos A \sin B$Various trigonometric sums

🎯 Worked Examples

Example 1: Arithmetic Sequence

Problem: Find the sum of the first 20 terms of the arithmetic sequence: 2, 5, 8, 11, …

Solution:

  1. First term: $a_1 = 2$
  2. Common difference: $d = 5 - 2 = 3$
  3. 20th term: $a_{20} = 2 + (20-1) \cdot 3 = 2 + 57 = 59$
  4. Sum: $S_{20} = \frac{20}{2}(2 + 59) = 10 \cdot 61 = 610$
  5. Answer: $610$

Example 2: Geometric Sequence

Problem: Find the sum of the first 6 terms of the geometric sequence: 2, 6, 18, 54, …

Solution:

  1. First term: $a_1 = 2$
  2. Common ratio: $r = \frac{6}{2} = 3$
  3. Sum: $S_6 = 2 \cdot \frac{1-3^6}{1-3} = 2 \cdot \frac{1-729}{-2} = 2 \cdot \frac{-728}{-2} = 728$
  4. Answer: $728$

Example 3: Telescoping Series

Problem: Find the sum $\sum_{k=1}^n \frac{1}{k(k+1)}$.

Solution:

  1. Partial fractions: $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$
  2. Write out terms: $\sum_{k=1}^n \left(\frac{1}{k} - \frac{1}{k+1}\right)$
  3. Telescope: $\left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right)$
  4. Cancel: $1 - \frac{1}{n+1} = \frac{n}{n+1}$
  5. Answer: $\frac{n}{n+1}$

⚠️ Common Pitfalls

Pitfall: Confusing arithmetic and geometric formulas

  • Fix: Arithmetic uses addition (difference), geometric uses multiplication (ratio)
  • Example: Arithmetic: $a_n = a_1 + (n-1)d$; Geometric: $a_n = a_1 \cdot r^{n-1}$

Pitfall: Forgetting to check if geometric series converges

  • Fix: Infinite geometric series only converges if $|r| < 1$
  • Example: $1 + 2 + 4 + 8 + \cdots$ diverges because $r = 2 > 1$

Pitfall: Off-by-one errors in term counting

  • Fix: $a_n$ is the $n$th term, so $a_1$ is the first term
  • Example: If $a_1 = 3$ and $d = 2$, then $a_5 = 3 + 4 \cdot 2 = 11$ (not $3 + 5 \cdot 2$)

🎯 AMC-Style Worked Example

Problem: Find the sum of the first 100 terms of the sequence: $1, 3, 6, 10, 15, 21, \ldots$

Solution:

  1. Recognize pattern: This is the sequence of triangular numbers
  2. Find formula: $a_n = \frac{n(n+1)}{2}$ (triangular number formula)
  3. Set up sum: $S_{100} = \sum_{n=1}^{100} \frac{n(n+1)}{2} = \frac{1}{2} \sum_{n=1}^{100} n(n+1)$
  4. Expand: $S_{100} = \frac{1}{2} \sum_{n=1}^{100} (n^2 + n) = \frac{1}{2} \left(\sum_{n=1}^{100} n^2 + \sum_{n=1}^{100} n\right)$
  5. Use formulas:
    • $\sum_{n=1}^{100} n = \frac{100 \cdot 101}{2} = 5050$
    • $\sum_{n=1}^{100} n^2 = \frac{100 \cdot 101 \cdot 201}{6} = 338350$
  6. Calculate: $S_{100} = \frac{1}{2}(338350 + 5050) = \frac{1}{2} \cdot 343400 = 171700$
  7. Answer: $171700$

Key insight: Some sequences have known formulas that can be used directly.

🔗 Related Patterns

  • Polynomials — Sequence formulas are often polynomials
  • Series — Sequences lead to series when summed
  • Telescoping — Special technique for certain series
  • Word Problems — Sequences often model real-world situations

📝 Practice Checklist

  • Master arithmetic sequence formulas
  • Practice geometric sequence formulas
  • Learn telescoping techniques
  • Practice sum problems
  • Understand convergence conditions
  • Work on speed and accuracy

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