Sequences & Series

Key concepts

• Arithmetic: Tₙ = a + (n−1)d
• Geometric: Tₙ = a·rⁿ⁻¹
• Sn arithmetic: n/2·(2a + (n−1)d)
• Sn geometric: a(rⁿ − 1)/(r − 1)
• S∞ = a/(1 − r) when |r| < 1

How it works

Decide whether the sequence is arithmetic (constant difference) or geometric (constant ratio). Use the correct nth-term and sum formulas. For sigma notation, identify the first term, last term and common difference/ratio. Identify the pattern before choosing a formula: a constant first difference means arithmetic, a constant ratio means geometric, and a constant second difference means quadratic. For sigma notation, write out the first few terms to find a and d or r, and count the terms as top value − bottom value + 1.

Step by step

1. Classify

Arithmetic, geometric, or neither?

2. Find a, d or r

First term and common difference/ratio.

3. Apply the formula

Use Tₙ or Sn as required.

4. Check n

When you solve for n, it must be a positive whole number; reject any other solution.

Worked examples

Find S∞ for 8 + 4 + 2 + 1 + …

1. r = 1/2, |r| < 1
2. S∞ = 8 / (1 − 1/2) = 16

Answer: S∞ = 16

Find T₂₀ of the sequence 5; 9; 13; …

1. Arithmetic: a = 5 and d = 4
2. T₂₀ = 5 + 19(4)

Answer: T₂₀ = 81

Find the general term of the quadratic sequence 3; 8; 15; 24; …

1. First differences: 5; 7; 9
2. Second difference: 2, so 2a = 2 and a = 1
3. 3a + b = 5, so b = 2
4. a + b + c = 3, so c = 0

Answer: Tₙ = n² + 2n

Calculate Σ (k = 1 to 10) of (3k − 1)

1. Arithmetic series with a = 2, d = 3 and n = 10
2. S₁₀ = 10/2 [2(2) + 9(3)] = 5 × 31

Answer: 155

How many terms of 3 + 7 + 11 + … add up to 210?

1. Sₙ = n/2 [2(3) + (n − 1)(4)] = n(2n + 1) = 210
2. 2n² + n − 210 = 0, so (2n + 21)(n − 10) = 0
3. n must be a positive whole number

Answer: n = 10

A geometric sequence has T₂ = 6 and T₅ = 162. Find a and r.

1. ar = 6 and ar⁴ = 162
2. Divide: r³ = 27, so r = 3
3. a = 6 ÷ 3

Answer: a = 2 and r = 3

Common mistakes

• Using arithmetic formula on a geometric sequence
• Applying S∞ when |r| ≥ 1 (diverges)
• Off-by-one errors with n
• Treating a quadratic sequence as arithmetic because the first differences look regular

Frequently asked questions

When does a geometric series converge?

When |r| < 1.

How do I count the terms in sigma notation?

Subtract the bottom value from the top value and add 1. For example, from k = 3 to k = 12 there are 10 terms.