A sequence is a list of numbers that follows a rule. The two kinds you will meet in algebra are arithmetic sequences, which add the same number each time, and geometric sequences, which multiply by the same number each time. Each kind has one formula for any term and one for the sum.
Telling them apart
Look at consecutive terms.
- Subtract each term from the next. If the difference is always the same, it is arithmetic. That difference is called d.
- Divide each term by the one before. If the ratio is always the same, it is geometric. That ratio is called r.
| Sequence | Test | Type |
|---|---|---|
| 5, 9, 13, 17, … | differences are all 4 | arithmetic, d = 4 |
| 3, 6, 12, 24, … | ratios are all 2 | geometric, r = 2 |
| 20, 14, 8, 2, … | differences are all −6 | arithmetic, d = −6 |
| 81, 27, 9, 3, … | ratios are all 1/3 | geometric, r = 1/3 |
| 1, 4, 9, 16, … | neither is constant | neither (these are squares) |
Arithmetic sequences behave like linear functions, with d as the slope. Geometric sequences behave like exponential functions, with r as the growth factor.
Arithmetic sequences
The nth term is:
aₙ = a₁ + (n − 1)d
a₁ is the first term. You add d one fewer time than the term number, because the first term has no d added yet.
Example Find the 40th term of 7, 11, 15, 19, …
- a₁ = 7 and d = 4.
- a₄₀ = 7 + (40 − 1)(4) = 7 + 156 = 163.
Example The 5th term of an arithmetic sequence is 23 and the 12th term is 58. Find the first term and d.
- From term 5 to term 12 is 7 steps, and the value rose 58 − 23 = 35.
- d = 35 ÷ 7 = 5.
- a₅ = a₁ + 4d, so 23 = a₁ + 20 and a₁ = 3.
The sum of an arithmetic sequence
To add the first n terms, average the first and last term and multiply by how many terms there are:
Sₙ = n(a₁ + aₙ)/2
Example Find the sum of the first 100 positive integers
n = 100, a₁ = 1, a₁₀₀ = 100. S = 100(1 + 100)/2 = 5,050.
This works because pairing the first and last terms, then the second and second-to-last, gives equal sums every time.
Example Find the sum 4 + 10 + 16 + … + 118
- d = 6. Find how many terms: 118 = 4 + (n − 1)(6), so n − 1 = 19 and n = 20.
- S = 20(4 + 118)/2 = 1,220.
Geometric sequences
The nth term is:
aₙ = a₁ × rn − 1
Example Find the 8th term of 3, 6, 12, 24, …
a₁ = 3 and r = 2. a₈ = 3 × 2⁷ = 3 × 128 = 384.
Example Find the 6th term of 200, −100, 50, −25, …
- r = −100/200 = −1/2. A negative ratio makes the signs alternate.
- a₆ = 200 × (−1/2)⁵ = 200 × (−1/32) = −6.25.
The sum of a geometric sequence
Sₙ = a₁(1 − rⁿ)/(1 − r), for r ≠ 1.
Example Find 2 + 6 + 18 + … to 7 terms
a₁ = 2, r = 3, n = 7. S₇ = 2(1 − 3⁷)/(1 − 3) = 2(1 − 2,187)/(−2) = 2,186.
Infinite geometric series
If −1 < r < 1, the terms shrink toward zero and the sum of infinitely many of them settles at a finite number:
S = a₁/(1 − r)
Example Find 12 + 6 + 3 + 1.5 + …
r = 1/2, so S = 12/(1 − 1/2) = 24.
If |r| ≥ 1 the terms don’t shrink and the infinite sum has no finite value.
Repeating decimals are geometric series
0.777… = 0.7 + 0.07 + 0.007 + … has a₁ = 0.7 and r = 0.1. The sum is 0.7/0.9 = 7/9.
Recursive formulas
A recursive formula gives the first term and a rule for getting each term from the one before.
- Arithmetic: a₁ = 7, aₙ = aₙ₋₁ + 4.
- Geometric: a₁ = 3, aₙ = 2 × aₙ₋₁.
Recursive rules are easy to read but slow for far-off terms. To find the 100th term, convert to the explicit formula above.
Your turn
Is 2, 5, 10, 17, 26, … arithmetic, geometric or neither?
Show answer
Neither. The differences are 3, 5, 7, 9, which change, and the ratios change too. (The terms are n² + 1.)
Find the 25th term of 50, 47, 44, 41, …
Show answer
−22. a₂₅ = 50 + 24(−3) = 50 − 72.
Find the sum of the first 30 terms of 5, 12, 19, 26, …
Show answer
3,195. d = 7, a₃₀ = 5 + 29(7) = 208, and S = 30(5 + 208)/2.
The 2nd term of a geometric sequence is 12 and the 5th term is 324. Find r and a₁.
Show answer
r = 3 and a₁ = 4. From term 2 to term 5 you multiply by r three times: r³ = 324/12 = 27.
Find the sum of the infinite series 45 + 15 + 5 + …
Show answer
67.5. r = 1/3, so S = 45/(2/3).
A ball is dropped from 10 m and each bounce reaches 80% of the previous height. How high is the 5th bounce?
Show answer
About 3.28 m. The bounces are 8, 6.4, 5.12, 4.096, 3.2768, which is 10(0.8)⁵.