Understanding patterns and mathematical progressions
A sequence is an ordered list of numbers that follow a specific pattern or rule.
2, 5, 8, 11, 14, ...
Pattern: Add 3 each time
a_n = 2 + 3(n-1)
3, 6, 12, 24, 48, ...
Pattern: Multiply by 2 each time
a_n = 3 \times 2^{n-1}
1, 1, 2, 3, 5, 8, 13, ...
Pattern: Each term is sum of previous two
a_n = a_{n-1} + a_{n-2}
Each term is found by adding a constant difference to the previous term.
a_n = a_1 + (n-1)d
Find the 15th term of the sequence: 5, 9, 13, 17, ...
Common difference d =
15th term =
Which term of 7, 11, 15, 19, ... is 99?
Each term is found by multiplying the previous term by a constant ratio.
a_n = a_1 \times r^{n-1}
Find the 8th term of: 2, 6, 18, 54, ...
Common ratio r =
8th term =
In 4, 12, 36, 108, ... which term is 78732?
A series is the sum of the terms of a sequence.
\sum_{i=1}^{n} a_i = a_1 + a_2 + ... + a_n
Read as: "Sum from i=1 to n of a_i"
S_n = \frac{n}{2} \times (a_1 + a_n)
or
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
S_n = a_1 \times \frac{1 - r^n}{1 - r}
(for r ≠ 1)
Investment growth: $1000 at 5% annual interest
A_n = 1000 \times (1.05)^n
Bacterial growth: doubles every hour
P_n = P_0 \times 2^n
Height decreases by 20% each bounce
h_n = h_0 \times (0.8)^n
You invest $2000 at 6% annual interest, compounded yearly. How much will you have after 10 years?
A_n = 2000 \times (1.06)^n
Amount after 10 years: $