Sequences & series
Find far-off terms and add up whole sequences.
Pen and paper is fine · no calculator needed why?
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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
- Find the change: second term minus first, or the ratio, second ÷ first.
- The nth term is n − 1 steps past the first.
- Arithmetic: first + (n − 1) × difference. Geometric: first × ratio to the power n − 1.
worked example
The sequence 50, 47, 44, … goes down by the same amount each time. What is the 25th term?
- The difference is 47 − 50 = −3.
- From the 1st term to the 25th is 24 steps.
- 50 + 24 × (−3) = 50 − 72 = −22.
Answer: −22
Pair the first and last
- Find the last term: first + (n − 1) × difference.
- Add the first and last terms. Every pair taken from the two ends has that same total.
- Multiply by n, then halve. Writing the list forward and backward makes n such pairs, twice the sum.
worked example
What is the sum of the first 10 terms of 4, 7, 10, …?
- Last term: 4 + 9 × 3 = 31.
- Sum = 10 × (4 + 31) ÷ 2 = 175.
Answer: 175
Divide by 1 − r
- Find the ratio r: second term ÷ first. Keep its sign.
- Work out 1 − r. A negative ratio makes it bigger than 1.
- Divide the first term by 1 − r.
worked example
What is the sum of the infinite geometric series 20 − 10 + 5 − …? Give a fraction or a whole number.
- The ratio is −10 ÷ 20 = −1/2.
- 1 − (−1/2) = 3/2.
- 20 ÷ (3/2) = 40/3.
Answer: 40/3
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.
practice
Arithmetic nth term
worked example
The sequence 4, 6, 8, … goes up by the same amount each time. What is the 46th term?
Answer: 94
- The difference is 6 − 4 = 2.
- aₙ = a₁ + (n − 1)d = 4 + 45 × 2 = 94.
Geometric nth term
worked example
The sequence 3, −6, 12, … is multiplied by −2 each time. What is the 5th term?
Answer: 48
- aₙ = a₁ × r^(n − 1) = 3 × (−2)⁴ = 3 × 16 = 48.
Arithmetic series
worked example
What is the sum of the first 16 terms of 6, 15, 24, …?
Answer: 1,176
- Last term: 6 + 15 × 9 = 141.
- Sum = 16 × (6 + 141) ÷ 2 = 1,176.
Infinite geometric sum
worked example
What is the sum of the infinite geometric series 38 + 57/2 + 171/8 + …? Give a fraction or a whole number.
Answer: 152
- The ratio is (57/2) ÷ 38 = 3/4.
- The sum is a ÷ (1 − r) = 38 ÷ (1 − 3/4) = 38 ÷ (1/4) = 152.