What is the difference between a sequence and a series?

A sequence is the ordered list of terms itself (3, 7, 11, 15, ...). A series is what you get when you add those terms together (3 + 7 + 11 + 15). Questions using the word 'sum', 'total', or sigma notation are asking about a series.

How do I know when an infinite geometric series has a sum?

Only when the common ratio r satisfies |r| < 1, that is, r is strictly between -1 and 1. Then the sum is a_1/(1 - r). If |r| is 1 or larger the partial sums grow without bound or oscillate, and there is no finite sum to report.

Is mathematical induction part of this topic?

It depends on the course. Many Grade 11-12 syllabuses pair induction with series because summation formulas are the standard practice problems for it. Tell the tutor whether your class covers induction and the sessions will include or skip it accordingly.

My child can do arithmetic sequences but freezes on sigma notation. Is that normal?

Yes — it is usually a reading problem, not a maths problem. Sigma notation compresses a start value, a stop value, and a rule into one symbol, and students often ignore the limits. The fix is to expand two or three sums by hand until the notation feels like shorthand for something already familiar.

Sequences & Series

Q: Is mathematical induction part of this topic?

It depends on the course. Many Grade 11-12 syllabuses pair induction with series because summation formulas are the standard practice problems for it. Tell the tutor whether your class covers induction and the sessions will include or skip it accordingly.

Q: My child can do arithmetic sequences but freezes on sigma notation. Is that normal?

Yes — it is usually a reading problem, not a maths problem. Sigma notation compresses a start value, a stop value, and a rule into one symbol, and students often ignore the limits. The fix is to expand two or three sums by hand until the notation feels like shorthand for something already familiar.

High School (11-12) · Mathematics

Sequences and series at the Grade 11-12 level ask you to describe a list of numbers with a formula, then add that list up — sometimes forever. The work splits into arithmetic and geometric patterns, sigma notation, closed-form sums, and deciding when an infinite geometric series has a finite total. Most problems are less about arithmetic and more about bookkeeping: which term is the first, how many terms are there, and which formula matches the pattern in front of you. Sessions are spoken, one-on-one, and built around working problems out loud so the tutor can hear where a step goes wrong rather than only seeing a final answer.

Start a session on Sequences & Series

What this covers

Writing a sequence both explicitly (a_n = a_1 + (n-1)d) and recursively, and converting between the two forms
Identifying arithmetic vs geometric patterns from a list of terms, and finding a missing term, common difference, or common ratio from two given terms
Reading and writing sigma notation, including shifting the index and splitting or combining sums
Using the partial sum formulas S_n = n/2(a_1 + a_n) and S_n = a_1(1 - r^n)/(1 - r), including solving backwards for n or r
Testing an infinite geometric series with |r| < 1 and finding its sum, including repeating decimals written as fractions
Proof by mathematical induction for summation formulas and divisibility statements, where the syllabus includes it

Where learners get stuck

Reporting the nth term when the question asked for the sum of n terms (or the reverse): a_n and S_n are introduced together and both use the same letters a_1, n, d and r, so students pattern-match on the symbols rather than on what the sentence asked for. Saying the answer out loud as a sentence — 'the 12th term is 47' vs 'the first 12 terms add to 312' — fixes it fast.
Believing any infinite series whose terms shrink toward zero must add up to a finite number: Every convergent example students meet early is geometric with a small ratio, so 'terms get tiny' becomes the remembered rule. The harmonic series 1 + 1/2 + 1/3 + ... is the standard counterexample; the actual test at this level is narrower — the series must be geometric and |r| must be less than 1.
Off-by-one errors in the number of terms, especially in sums like the terms from the 7th to the 20th: Students subtract 20 - 7 and get 13 instead of 14, or start a sigma index at 0 while using a formula written for a starting index of 1. It comes from treating n as a position label and a count interchangeably; counting a tiny case by hand first is the reliable check.

What a session looks like

A typical session opens with the tutor asking you to describe a given sequence in words before any formula appears, so the pattern type is settled first. You then work through problems by voice — stating the first term, the difference or ratio, and which formula you have chosen and why — while the tutor questions the steps that are ambiguous rather than supplying the next line. Later parts of a session usually push toward the harder variants: finding n when the sum is given, handling sigma notation with a shifted lower limit, or writing out the induction step in full. The tutor adapts to the notation your class uses (a_1 vs u_1, t_n vs a_n) if you say which one.

Helpful to know first

Comfort with algebraic manipulation: solving linear equations and rearranging a formula for any of its variables
Exponent rules, including r^n and simplifying expressions like r^(n-1) · r
Solving a pair of simultaneous equations, used constantly to find a_1 and d from two given terms
Basic function notation and the idea of an input n producing an output a_n
For geometric problems where n is the unknown, some exposure to logarithms is helpful but the tutor can work around it

Questions

What is the difference between a sequence and a series?: A sequence is the ordered list of terms itself (3, 7, 11, 15, ...). A series is what you get when you add those terms together (3 + 7 + 11 + 15). Questions using the word 'sum', 'total', or sigma notation are asking about a series.
How do I know when an infinite geometric series has a sum?: Only when the common ratio r satisfies |r| < 1, that is, r is strictly between -1 and 1. Then the sum is a_1/(1 - r). If |r| is 1 or larger the partial sums grow without bound or oscillate, and there is no finite sum to report.
Is mathematical induction part of this topic?: It depends on the course. Many Grade 11-12 syllabuses pair induction with series because summation formulas are the standard practice problems for it. Tell the tutor whether your class covers induction and the sessions will include or skip it accordingly.
My child can do arithmetic sequences but freezes on sigma notation. Is that normal?: Yes — it is usually a reading problem, not a maths problem. Sigma notation compresses a start value, a stop value, and a rule into one symbol, and students often ignore the limits. The fix is to expand two or three sums by hand until the notation feels like shorthand for something already familiar.