Sequences & Series
Start gently
A sequence is a list of numbers whose rule you find, so you can write the n-th term as a formula. An arithmetic sequence grows by a fixed amount; a geometric one grows by a fixed multiple.
Key points
Arithmetic Sequences
A sequence in which the gap between any two neighboring terms is always the same. That gap d is called the common difference, and the nth term is a_n = a_1 + (n−1)d.
Geometric Sequences
A sequence in which the ratio between any two neighboring terms is always the same. That ratio r is called the common ratio, and the nth term is a_n = a_1 × r^{n−1}.
Summing an Arithmetic Sequence
For an arithmetic sequence with first term a_1, last term l, and n terms in total, the sum is S_n = n(a_1 + l)/2 — the number of terms times the average of the first and last one.
Summing a Geometric Sequence
For a geometric sequence with first term a_1, common ratio r (with r ≠ 1), and n terms in total, the sum is S_n = a_1(1−r^n)/(1−r).
Where You'll Meet This: Compound Interest
Start with an amount P earning interest r per year. The balance at the end of each year forms a geometric sequence, so after n years you have P × (1+r)^n. Leave ¥1,000,000 at 5% for 10 years and it grows to roughly ¥1,630,000.
See it drawn
In the arithmetic sequence 2, 5, 8, 11, 14 the tops of the bars line up straight.
This one accelerates. Compound interest and the spread of an infection have exactly this shape.
Jobs that use this
Financial Analyst$160k
Growth Engineer (Startup)$140k
Algorithms Engineer$170k
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