Sequences and Series
Grade 11 · Math · Study Guide
Understand arithmetic and geometric sequences, partial sums, and when infinite geometric series converge.
An arithmetic sequence adds a constant difference d between consecutive terms: aₙ = a₁ + (n−1)d. A geometric sequence multiplies by a constant ratio r: aₙ = a₁ · rⁿ⁻¹.
A series is the sum of terms in a sequence. The partial sum Sₙ for an arithmetic series is Sₙ = n(a₁ + aₙ)/2. For a geometric series: Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1.
An infinite geometric series converges to S = a₁/(1 − r) when |r| < 1 and diverges when |r| ≥ 1.
| Type | nth Term | Partial Sum Sₙ |
|---|---|---|
| Arithmetic | aₙ = a₁ + (n−1)d | Sₙ = n(a₁ + aₙ)/2 |
| Geometric | aₙ = a₁ · rⁿ⁻¹ | Sₙ = a₁(1 − rⁿ)/(1 − r) |
| Condition | Result |
|---|---|
| |r| < 1 | Converges; S = a₁/(1 − r) |
| |r| ≥ 1 | Diverges (sum is infinite) |
Key terms
- Arithmetic sequence
- A sequence with a constant difference d between consecutive terms.
- Geometric sequence
- A sequence with a constant ratio r between consecutive terms.
- Partial sum (Sₙ)
- The sum of the first n terms of a sequence.
- Sigma notation
- Compact notation for a sum: Σᵢ₌₁ⁿ aᵢ means a₁ + a₂ + … + aₙ.
Practice questions
Find the 10th term of the arithmetic sequence 3, 7, 11, 15, …
- 39
- 40
- 41
- 43
Show answer
d = 4, a₁ = 3. a₁₀ = 3 + (10−1)·4 = 3 + 36 = 39.
Find the sum of the first 8 terms of the arithmetic series 5 + 8 + 11 + …
Show answer
a₈ = 5 + 7×3 = 26. S₈ = 8(5+26)/2 = 8×31/2 = 124.
In the geometric sequence 2, 6, 18, 54, …, what is the 5th term?
- 108
- 162
- 486
- 162
Show answer
r = 3, a₅ = 2 · 3⁴ = 2 · 81 = 162.
Does the infinite series 4 + 2 + 1 + 1/2 + … converge? If so, to what value?
- Diverges
- 8
- 7
- Converges to 6
Show answer
r = 1/2, |r| < 1. S = a₁/(1−r) = 4/(1−1/2) = 4/(1/2) = 8.
Write in expanded form: Σᵢ₌₁⁴ (2i + 1).
Show answer
i=1: 3; i=2: 5; i=3: 7; i=4: 9. Sum = 24.
Fun facts
- Zeno's paradox of Achilles and the tortoise is resolved by an infinite geometric series: 1 + 1/2 + 1/4 + … converges to 2.
- The total distance a bouncing ball travels (up and down) after being dropped can be computed with an infinite geometric series.
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