Series: arithmetic and geometric progressions, sums, binomial expansion

Series: Arithmetic & Geometric Progressions, Sums & Binomial Expansion

Arithmetic Progressions (AP)

Definition: A sequence where each term after the first is found by adding a constant difference d to the previous term.
$a_n = a_1 + (n-1)d$

Analogy: Think of a line of friends standing 2 m apart. If the first friend is at 0 m, the second is at 2 m, the third at 4 m, and so on. The distance between each friend is the common difference d = 2 m.

Sum of the first n terms

Two handy formulas:
$$S_n = \frac{n}{2}\,(a_1 + a_n)$$
$$S_n = \frac{n}{2}\,[\,2a_1 + (n-1)d\,]$$

Example: Find the sum of the first 10 terms of the AP 3, 7, 11, …
Here, $a_1 = 3$, $d = 4$, $n = 10$, $a_{10} = 3 + 9\cdot4 = 39$.
$S_{10} = \frac{10}{2}\,(3 + 39) = 5 \times 42 = 210$.

Exam Tip: Always check if the problem asks for the sum of the first n terms or the sum of a specific range. If it gives two terms (e.g., 3rd and 8th), use $S_n = \frac{n}{2}\,(a_1 + a_n)$ with $n$ equal to the number of terms in that range.

Geometric Progressions (GP)

Definition: A sequence where each term after the first is found by multiplying the previous term by a constant ratio r.
$a_n = a_1 r^{\,n-1}$

Analogy: Imagine a snowball rolling down a hill, doubling its size each step. If the first snowball is 1 cm, the second is 2 cm, the third 4 cm, etc. Here, $r = 2$.

Sum of the first n terms

$$S_n = a_1\,\frac{1-r^n}{1-r}\quad (r eq 1)$$

Example: Sum the first 5 terms of the GP 2, 6, 18, …
$a_1 = 2$, $r = 3$, $n = 5$.
$S_5 = 2\,\frac{1-3^5}{1-3} = 2\,\frac{1-243}{-2} = 2\,\frac{-242}{-2} = 242$.

Exam Tip: For a GP where $r > 1$, the sum grows quickly. If you’re asked for the sum of a large number of terms, check if the problem allows you to use the formula for an infinite GP: $S_\infty = \frac{a_1}{1-r}$ (only valid if $|r|<1$).

Binomial Expansion

Binomial Theorem:
$$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{\,n-k} y^k$$

Binomial coefficient: $\displaystyle \binom{n}{k} = \frac{n!}{k!\,(n-k)!}$
Think of it as “n choose k” – the number of ways to pick k items from n.

Expanding a Binomial

Example 1: Expand $(x+2)^3$.
$$\begin{aligned} (x+2)^3 &= \binom{3}{0}x^3 2^0 + \binom{3}{1}x^2 2^1 + \binom{3}{2}x^1 2^2 + \binom{3}{3}x^0 2^3\\ &= 1\cdot x^3 + 3\cdot x^2\cdot 2 + 3\cdot x\cdot 4 + 1\cdot 8\\ &= x^3 + 6x^2 + 12x + 8 \end{aligned}$$

Example 2: Expand $(2x-3)^4$ (note the minus sign).
$$\begin{aligned} (2x-3)^4 &= \binom{4}{0}(2x)^4(-3)^0 + \binom{4}{1}(2x)^3(-3)^1 + \binom{4}{2}(2x)^2(-3)^2\\ &\quad + \binom{4}{3}(2x)^1(-3)^3 + \binom{4}{4}(2x)^0(-3)^4\\ &= 16x^4 - 96x^3 + 216x^2 - 216x + 81 \end{aligned}$$

Exam Tip: Memorise the first few rows of Pascal’s Triangle (1; 1 2; 1 3 3 1; 1 4 6 4 1). They give the binomial coefficients for $n = 1, 2, 3, 4$. For larger $n$, use the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ or simplify step‑by‑step using the pattern.

Final Exam Reminder: Practice deriving sums from scratch, not just plugging into formulas. Write out the first few terms, identify $a_1$, $d$ or $r$, and then apply the appropriate sum formula. For binomial expansions, always keep track of signs and powers carefully. Good luck! 🚀