doing math

courses › Algebra

Sequences & series

level 29 course

Find far-off terms and add up whole sequences.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 29.

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Builds on: Linear equations (not open yet) · Exponents & roots (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

An arithmetic sequence adds the same amount each time: 7, 11, 15 adds 4. A geometric sequence multiplies by the same amount: 3, 6, 12 doubles.

Count the steps with care. From the 1st term to the 20th there are 19 steps, because the 1st term is there before any step.

A series is the sum of the terms. An endless geometric series has a finite total when its ratio r is between −1 and 1. After the first term a, the rest is r times the whole series, which is why the total is a ÷ (1 − r).

Techniques

Count the steps

You want the nth term of an arithmetic or geometric sequence.

  1. Find the change: second term minus first, or the ratio, second ÷ first.
  2. The nth term is n − 1 steps past the first.
  3. Arithmetic: first + (n − 1) × difference. Geometric: first × ratio to the power n − 1.
worked example

Example: The sequence 50, 47, 44, … goes down by the same amount each time. What is the 25th term?

  1. The difference is 47 − 50 = −3.
  2. From the 1st term to the 25th is 24 steps.
  3. 50 + 24 × (−3) = 50 − 72 = −22.

Answer: −22

Pair the first and last

Adding the first n terms of an arithmetic sequence.

  1. Find the last term: first + (n − 1) × difference.
  2. Add the first and last terms. Every pair taken from the two ends has that same total.
  3. Multiply by n, then halve. Writing the list forward and backward makes n such pairs, twice the sum.
worked example

Example: What is the sum of the first 10 terms of 4, 7, 10, …?

  1. Last term: 4 + 9 × 3 = 31.
  2. Sum = 10 × (4 + 31) ÷ 2 = 175.

Answer: 175

Divide by 1 − r

An endless geometric series with a ratio between −1 and 1.

  1. Find the ratio r: second term ÷ first. Keep its sign.
  2. Work out 1 − r. A negative ratio makes it bigger than 1.
  3. Divide the first term by 1 − r.
worked example

Example: What is the sum of the infinite geometric series 20 − 10 + 5 − …? Give a fraction or a whole number.

  1. The ratio is −10 ÷ 20 = −1/2.
  2. 1 − (−1/2) = 3/2.
  3. 20 ÷ (3/2) = 40/3.

Answer: 40/3

Tips by skill

  • TipArithmetic nth term: Find the difference, sign included. The nth term is the first term plus n − 1 differences.
  • TipGeometric nth term: Find the ratio. The nth term is the first term times the ratio to the power n − 1. Watch the sign.
  • TipArithmetic series: Find the last term, add it to the first, multiply by the number of terms, then halve.
  • TipInfinite geometric sum: Find the ratio r, second term ÷ first, sign included. The sum is the first term ÷ (1 − r).

Watch out for

  • Taking n steps instead of n − 1.
  • Losing the sign with a ratio of −2. An odd power of −2 is negative; an even power is positive.
  • Dividing by r instead of by 1 − r in an endless sum.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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