Series
The binomial expansion
- For a positive integer $n$:
$$(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + b^n$$
- where $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$ is a binomial coefficient.
Practice
What is the binomial coefficient C(5, 2) = 5! / (2! 3!)?
C(5,2) = 120/(2×6) = 120/12 = 10.
AP and GP
- Arithmetic progression (AP) — add a common difference $d$:
- $u_n = a + (n-1)d$, $\quad S_n = \tfrac{n}{2}\big(2a + (n-1)d\big)$.
- Geometric progression (GP) — multiply by a common ratio $r$:
- $u_n = ar^{n-1}$, $\quad S_n = \dfrac{a(1-r^n)}{1-r}$.
- A GP converges when $|r| < 1$, with sum to infinity $S_\infty = \dfrac{a}{1-r}$.
Practice
An AP has first term a = 3 and common difference d = 4. What is the 5th term?
u₅ = a + (n−1)d = 3 + 4×4 = 3 + 16 = 19.
Practice
An AP has a = 2 and d = 3. What is the sum of the first 5 terms?
S₅ = (5/2)(2×2 + 4×3) = (5/2)(4 + 12) = (5/2)(16) = 40.
Practice
A GP has a = 8 and r = 0.5. What is the sum to infinity?
S∞ = a/(1−r) = 8/(1−0.5) = 8/0.5 = 16.
You've got it
Key idea
- AP: $u_n = a + (n-1)d$, $S_n = \tfrac{n}{2}(2a+(n-1)d)$
- GP: $u_n = ar^{n-1}$, $S_n = \dfrac{a(1-r^n)}{1-r}$
- if $|r|<1$, sum to infinity $S_\infty = \dfrac{a}{1-r}$