Sequences and series
| English | Chinese | Pinyin |
|---|---|---|
| arithmetic sequence | 等差数列 | děng chā shù liè |
| geometric sequence | 等比数列 | děng bǐ shù liè |
| sequence | 数列 | shù liè |
| series | 级数 | jí shù |
| common difference | 公差 | gōng chāi |
| common ratio | 公比 | gōng bǐ |
| sum to infinity | 无穷和 | wú qióng hé |
Money multiplies. It does not add
- Save 100 yuan a month and your savings grow by a fixed amount. Invest at 4% and they grow by a fixed factor.
- The first is an arithmetic sequence 等差数列; the second is a geometric sequence 等比数列.
- Nearly every financial question in GAC010 is geometric, which is why this unit comes first.
An investment grows by 4% each year. Which family is it?
A percentage is a factor, not an amount, so each year multiplies by 1.04.
The two sequences
- A sequence 数列 is an ordered list; a series 级数 is its sum.
- Arithmetic adds a common difference 公差 $d$: $u_n = a + (n-1)d$.
- Geometric multiplies by a common ratio 公比 $r$: $u_n = ar^{\,n-1}$.
- The $n-1$ in both is the trap. The first term has already happened, so after $n$ steps you are at term $n+1$.
Arithmetic against geometric
A constant factor eventually overtakes a constant amount, however large the amount.
An arithmetic sequence starts at 5 with common difference 3. What is the 10th term?
u₁₀ = 5 + 9 × 3 = 32. Nine steps from the first term, not ten.
Summing them
- A geometric series with $|r| < 1$ converges to a sum to infinity 无穷和 $S_\infty = \dfrac{a}{1-r}$.
- That only works when the terms shrink. With $|r| \ge 1$ the sum grows without limit and the formula is meaningless.
A geometric series has a sum to infinity only when...
The terms must shrink towards zero. With |r| ≥ 1 the sum grows without limit.
A machine worth 80,000 yuan loses 12% of its value each year. What is it worth after 5 years?
The ratio is $0.88$, and after 5 years the ratio has been applied five times:
In the formula this is term $u_6$, because $u_1$ is the value today. Getting that index right is the whole difficulty of the unit.
A machine worth 80000 loses 12% a year. What is it worth after 5 years, to the nearest 100?
80000 × 0.88⁵ = 42200. Multiplying by 0.88 five times, not subtracting 12% five times.
Ask whether the quantity changes by an amount or by a factor. Amount means arithmetic; factor means geometric. Every financial question in this module answers "factor", and choosing the wrong family makes every later line wrong.
$0.88^5$ is not "88% five times taken away". It is the value multiplied by 0.88 five times, which leaves 52.8%, not 40%. Subtracting 12% five times is the standard error, and it gives an answer that is wrong by thousands.
Losing 12% a year for five years means losing 60% in total.
0.88⁵ = 0.528, so about 47% is lost. Each year's loss applies to a smaller amount than the last.