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Sequences and the General Term
A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.
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Sequences and the General Term
What is it?
A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.
In simple words
- A sequence is a list a1, a2, a3, ...
- with a rule for the nth term a_n.
- Finite: stops.
- an is one term. Sn is a sum. Do not call S_n a sequence term.
Important terms
SEQUENCES AND THE GENERAL TERM: A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.
1. SEQUENCE: Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a{n+1} from a_n (later AP/GP).
2. SERIES: Sn = a1+...+an. Sigma notation named: sumk=1n ak. Do not mix an (one term) with S_n (a total).
What you should learn
- State the main ideas and notation used in sequences and the general term.
- Apply the rules of sequences and the general term in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- a1 first term
- Write 4 terms
- n is a positive integer
SEQUENCE
Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a{n+1} from a_n (later AP/GP).
SERIES
Sn = a1+...+an. Sigma notation named: sumk=1n ak. Do not mix an (one term) with S_n (a total).
Recommended for Class 11
Full school-level explanation
A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.
List vs sum, then a_n:
WHAT — SEQUENCES AND THE GENERAL TERM
A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.
RULE — 1. SEQUENCE
Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a{n+1} from a_n (later AP/GP).
METHOD — 2. SERIES
Sn = a1+...+an. Sigma notation named: sumk=1n ak. Do not mix an (one term) with S_n (a total).
WORKED EXAMPLE
Worked: first four terms of a_n = n^2 - 1. n=1: 0 n=2: 3 n=3: 8 n=4: 15
SEQUENCE
Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a{n+1} from a_n (later AP/GP).
Remember
- nth term
Examples
- a1 first term
- Write 4 terms
- n is a positive integer
SERIES
Sn = a1+...+an. Sigma notation named: sumk=1n ak. Do not mix an (one term) with S_n (a total).
Remember
- S_n is a sum
Examples
- S1 = a1
- Sn - Sn-1 = an
- Infinite sum later GP
How to answer a sequences and the general term question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- sequence = list
- series = sum of the list
- 0,3,8,15
- term vs sum
- nth term
- S_n is a sum
- sequence = list
- series = sum of the list
How the ideas connect
Worked: first four terms of a_n = n^2 - 1. n=1: 0 n=2: 3 n=3: 8 n=4: 15
Deeper understanding
Go beyond memorising Sequences and the General Term. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: first four terms of a_n = n^2 - 1. n=1: 0 n=2: 3 n=3: 8 n=4: 15
Common mistakes
Exam focus
Core Idea
1. SEQUENCE
2. SERIES
Worked Example
Exam focus
Quick revision
- sequence = list
- series = sum of the list
- 0,3,8,15
- term vs sum
- SEQUENCES AND THE GENERAL TERM: A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes
- 1. SEQUENCE: Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a
an is one term. Sn is a sum. Do not call S_n a sequence term.
For long answers on Sequences and the General Term: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sequence = list
- series = sum of the list
- 0,3,8,15
- term vs sum
Important formulas
- sequence = list
- series = sum of the list
- 0,3,8,15
Common exam traps
- an is one term. Sn is a sum. Do not call S_n a sequence term.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
an is one term. Sn is a sum. Do not call S_n a sequence term.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
an is one term. Sn is a sum. Do not call S_n a sequence term.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sequence = list
series = sum of the list
0,3,8,15
term vs sum
A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meani
sequence = list
- Definition
- A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meani
- Formula
- sequence = list
- Example
- a1 first term
- Critical warning
- an is one term. Sn is a sum. Do not call S_n a sequence term.
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Exam Practice
Define
Q1. Define Sequences and the General Term.
Explain
Q2. Explain: SEQUENCE.
Example
Q3. Give one clear example of Sequences and the General Term and say why it fits.
Mistake
Q4. State one common mistake students make about Sequences and the General Term.
Exam note
Q5. Write a short exam note on Sequences and the General Term: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.