Porisma

Sequences and series

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By the end, you can

  • Distinguish a term from a sum
  • Use arithmetic, geometric and sigma formulas
  • Check convergence and analyse recurrences

about 2 hours for the lesson · practise at your own pace.

The central idea

A sequence lists terms; a series adds them. For arithmetic terms, un=a+(n−1)du_n=a+(n-1)d and Sn=n(2a+(n−1)d)/2S_n=n(2a+(n-1)d)/2. For geometric terms, un=arn−1u_n=ar^{n-1} and Sn=a(1−rn)/(1−r)S_n=a(1-r^n)/(1-r) for r≠1r\ne1. An infinite geometric sum is a/(1−r)a/(1-r) only when ∣r∣<1|r|<1. Recurrences define each new term from earlier ones.

Terms, indices and partial sums

The index n in u1,u2,…,un,…u_1,u_2,\ldots,u_n,\ldots records position. A series adds terms, and Sn=u1+⋯+unS_n=u_1+\cdots+u_n is the sum of its first n terms. The teacher note stresses that n is a positive integer.

An explicit formula un=f(n)u_n=f(n) treats position as input and the term as output. For un=n2+3u_n=n^2+3, the sequence starts 4,7,12,19,28,…4,7,12,19,28,\ldots and u100=10003u_{100}=10003. Restrict a finite sequence’s indices to its listed positions.

Explore

Use the worked example below, then change one input or assumption. Predict which part of the answer must change before calculating.

Related interactive example

Explore this related example, then return to the prediction task above.

Terms and running total for x = 0.8

Drag the slider to change n and compare the plotted values.

Indexn = 5
Reset the controls
term uₙ at n = 5
0.0032
sum Sₙ at n = 5
0.2062
Key idea

With ratio r=0.4r=0.4 the terms shrink geometrically and the partial sums level off at S∞=a1−rS_\infty=\frac{a}{1-r}.

Ideas to try
Slide to a large nn: the running total settles at S∞=0.412(0.8)(0.2)≈0.208S_\infty=\frac{0.4}{12(0.8)(0.2)}\approx0.208.
Open the full interactive problem →

Worked example

For 3,6,12,…3,6,12,\ldots, a=3a=3 and r=2r=2. The fourth term is 24, while the first four terms sum to 3(1−24)/(1−2)=453(1-2^4)/(1-2)=45. There is no finite sum to infinity.

A recurrence approaching a fixed point needs a stability argument; solving the fixed-point equation alone is not enough.

Check your understanding

For un=5(−1/2)n−1u_n=5(-1/2)^{n-1}, find the limiting term and the sum to infinity. Explain why these are different quantities. What happens to the partial sums if every term is 5 instead?

Practice questions

Practice section 1 · Questions 1–6

Generate terms and describe a formula · Separate monotonic behaviour from convergence · +4 more

Practice section 2 · Questions 7–12

Read an arithmetic progression and form its nth term · Find two specified arithmetic terms · +4 more

Practice section 3 · Questions 13–18

Recognise a common ratio · Form the nth term of a geometric progression · +4 more

Practice section 4 · Questions 19–24

Check a GP proof and a constant sequence · Recover terms and check the exceptional first term · +4 more

Practice section 5 · Questions 25–30

Evaluate three finite sums · Find the first harmonic partial sum above four · +4 more

Practice section 6 · Questions 31–36

Sum twenty arithmetic terms · Recover an AP from two consecutive blocks of sums · +4 more

Practice section 7 · Questions 37–42

Total the doubling reward on a chessboard · Evaluate a finite geometric sum above ratio one · +4 more

Practice section 8 · Questions 43–48

Resolve the constant cases left out of the table · Distinguish term limits from series limits · +4 more

Practice section 9 · Questions 49–54

Split an infinite GP into equal prefix and tail sums · Combine polynomial and geometric finite sums · +4 more

Practice section 10 · Questions 55–60

Compare AP and GP travel totals in a chase · Count travelled legs in a bouncing-ball model · +4 more

Practice section 11 · Questions 61–66

Exclude a digit using arithmetic sums · Sum increasing interest-free instalments · +4 more

Practice section 12 · Questions 67–72

Split mixed sigma sums into arithmetic and geometric parts · Change a summation range with two partial sums · +4 more

Practice section 13 · Questions 73–78

Use a logarithmic fixed-point graph to compare terms · Compare geometric and arithmetic instrument bars · +4 more

Practice section 14 · Questions 79–84

Match the final odd square to its summation index · Subtract the correct prefix from a finite sum · +4 more

Practice section 15 · Questions 85–90

Compare positive denominators to prove a strict lower bound · Sum the rows of a triangular labelled board · +4 more

Practice section 16 · Questions 91–96

Use an iteration to approximate the specified cubic root · Unfold a loss-then-top-up recurrence in two directions · +4 more

Practice section 17 · Questions 97–98

Remove grain before growth and retain the exact supply bound · Add a late fee after interest and count the charging cycles

More practice

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