Maths › Sequences and series › Sequences and sigma notation
Sequences and sigma notation
A sequence is a list with a law attached, given either as a formula for the nth term or as a recurrence that builds each term from the one before. Three words cover how such lists behave. One Greek letter compresses any sum of their terms into a single readable instruction.
Builds on Functions, inverses and the modulus.
IN THIS TOPIC
- Generate a sequence from an nth-term formula or from a recurrence, and classify how it behaves.
- Read and write sums in sigma notation.
COMMON MISCONCEPTION
A decreasing sequence must eventually go below zero.
Lists, and how they behave
An nth-term formula such as un = 1/(3n + 1) will give you any term you ask for, straight away. A recurrence such as un+1 = 3un − 2 works differently, building the list one step at a time from its predecessor, and it is worthless without a stated starting value. The paper always supplies that value. Use it.
WORKED EXAMPLE
Running a recurrence
A sequence has u1 = 3 and un+1 = 3un − 2. Find the first four terms.
u2 = 3 × 3 − 2 = 7, then u3 = 3 × 7 − 2 = 19, then u4 = 3 × 19 − 2 = 55.
The list runs 3, 7, 19, 55, each term roughly tripling.
Recurrence marks sit on the substitutions, so write every substitution down. A column of four bare numbers scores less than the same numbers with their working.
A sequence increases when un+1 > un at every step, and decreases when the inequality runs the other way. It is periodic if it repeats a fixed cycle, and the length of that cycle is called the order. The specification's own examples are 2n climbing, 1/(3n + 1) sinking, and un+1 = 1/un bouncing with period 2.
Decreasing does not mean unbounded below. Take un = 1 + 1/n. It falls at every step and still never reaches 1, let alone zero. Falling, and falling through the floor, are separate claims.
GUIDED PRACTICE
Classify with reasons
For un+1 = 1/un with u1 = 5, write down the first five terms and classify the sequence, before opening the working.
Show the working
The terms run 5, 1/5, 5, 1/5, 5.
Applying the rule twice returns the starting value, so the sequence is periodic with order 2.
“Periodic” on its own is half an answer. The order is the other half, and it carries its own mark.
Sigma notation
Sigma notation packs a sum into three pieces. Σ means add, the expression after it is the recipe for one term, and the counter runs from the number below to the number above.
One tiny result is worth memorising. Adding the constant 1 once per term across n terms gives Σ1 = n, which the arithmetic series lesson leans on within the week.
INDEPENDENT PRACTICE
Both directions at once
Evaluate the sum of (2r + 1) for r from 1 to 5, and write the sum 4 + 9 + 14 + 19 + 24 in sigma notation.
Show the working
The terms are 3, 5, 7, 9, 11, totalling 35.
The second list climbs in steps of 5 from 4, so term r is 5r − 1 and the sum is Σ(5r − 1) for r = 1 to 5.
Reading sigma and writing sigma are one skill seen from two sides, and papers ask for both.
ASSESSMENT FOCUS
- A recurrence without its first term defines nothing. Write u1 down before you start.
- Classify with the comparison visible. Give un+1 − un and its sign, or exhibit the cycle, and for a periodic sequence state the order as well as the word.
- Identify the counter's range and the term recipe separately before evaluating any sigma. For five or six terms, listing them is quicker than finding a formula.
CHECK YOURSELF
A sequence has un = 40 − 3n. Find u10, determine when terms first go negative, and evaluate Σun for n from 1 to 4.
Show a hint
Linear in n throughout; list four terms for the sum.
Show the answer
u10 = 40 − 30 = 10.
Terms go negative once 40 − 3n < 0, so from n = 14 onward, u14 being −2.
The first four terms are 37, 34, 31, 28, so the sum is 130.
A formula serves any term on demand; a recurrence builds the list step by step from a stated start.
Increasing, decreasing or periodic with its order, and Σ carries a counter, a recipe and a range.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the sequences and sigma notation questions page.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- Generate a sequence from an nth-term formula or from a recurrence, and classify how it behaves.
- Read and write sums in sigma notation.
Open the full revision checklist to see every objective in the course in one place.