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Sequences and sigma notation

A sequence is a list with a law, given either as a formula for the nth term or as a recurrence that builds each term from the last. Three words describe how lists behave, increasing, decreasing, periodic, and one Greek letter compresses any sum of terms into a single readable instruction.

Year 12-13EDEXCEL 9MA0 4.2, 4.3

Builds on Functions, inverses and the modulus.

IN THIS TOPIC

  • Generate sequences from nth-term formulae and from recurrence relations.
  • Classify sequences as increasing, decreasing or periodic, with justification.
  • Read and write sums in sigma notation.

WHAT YOU PROBABLY THINK

A decreasing sequence must eventually go below zero.

Two ways to give a list

An nth-term formula like un = 1/(3n + 1) hands over any term on demand; a recurrence like un+1 = 3un − 2 with u1 = 3 builds the list one step at a time, each term from its predecessor. The recurrence needs its starting value stated, or the machine has nothing to chew on.

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 questions are marked on the substitutions, so show each feed of the machine rather than only the outputs.

Three behaviours

A sequence is increasing if every term beats the one before, un+1 > un, decreasing if the inequality runs the other way, and periodic if it repeats a fixed cycle, the cycle's length being the order. The spec's own trio: 2n increases, 1/(3n + 1) decreases, and un+1 = 1/un bounces with period 2.

Three sequences plotted term by term: 2 to the n climbing, 1 over 3n plus 1 sinking towards zero, and the reciprocal recurrence bouncing between 5 and one fifth with period 2increasing: 2ⁿdecreasing: 1/(3n + 1)periodicthree behaviours, one vocabulary
FIG. 1The three behaviours, plotted: powers of 2 climbing, 1/(3n + 1) sinking toward zero, and the reciprocal recurrence alternating between 5 and 1/5 forever.

Decreasing does not mean unbounded, which retires the opening lie: un = 1 + 1/n falls with every step yet never reaches 1, let alone zero. A decreasing sequence needs only to fall, not to fall through the floor.

YOUR TURN

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” wants the order stated; the two-word classification is only half the answer.

Sigma notation

Sums of terms compress into sigma notation: the Σ says add, the expression after it is the recipe for each term, and the counter runs from the value below to the value above.

The anatomy of sigma notation: the sigma says add, the expression after it is the recipe for each term, the letter underneath is the counter, and the numbers below and above are where the counter starts and stopsΣr = 15(2r + 1)= 35stop valuestart valuethe recipe for term r3 + 5 + 7 + 9 + 11: five terms, added
FIG. 2The notation, labelled. Counter from 1 to 5, recipe 2r + 1, total 35: five odd numbers, added in one symbol.

One tiny sum is worth memorising: adding the constant 1 once per term gives Σ1 = n over n terms, a fact the arithmetic series lesson leans on shortly.

TRY IT UNSEEN

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 the same skill mirrored, and papers ask for both directions.

THE EXAM BIT

  • State the starting value with any recurrence; without it the sequence is not defined and the examiner knows it.
  • Classify with the comparison shown: un+1 − un positive, negative, or a cycle exhibited, then the word and the order.
  • Decreasing means falling, not falling forever; check for a floor before claiming the sequence passes any value.
  • In sigma notation identify the counter's range and the term recipe separately before evaluating anything.
  • Small sums evaluate fastest by listing the terms; save formulae for the series lessons where they belong.

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 when 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 compresses a sum into a counter, a recipe and a range.

CHECK YOUR PROGRESS

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  • Generate sequences from nth-term formulae and from recurrence relations.
  • Classify sequences as increasing, decreasing or periodic, with justification.
  • Read and write sums in sigma notation.

No animated video for this topic yet; these notes stand alone.