Active Maths 4 Chapter 11: Proof by Induction and Series Solutions

Vocabulary and formula flashcards covering Chapter 11 solutions on mathematical induction, series summation formulas, divisibility proofs, and inequalities.

Mathematical Induction

A formal method of mathematical proof used to demonstrate that a statement $P(n)$ is true for all natural numbers $n$, consisting of a base step, an inductive assumption, and an inductive step.

Base Step

The initial step in a proof by induction that verifies the proposition $P(n)$ is true for the starting value (usually $n = 1$ or another given initial value).

Inductive Assumption

The step in a proof by induction where the statement $P(k)$ is assumed to be true for an arbitrary positive integer $k$.

Inductive Step

The step in a proof by induction where it is proved algebraically that if the proposition $P(k)$ holds, then $P(k + 1)$ must also be true.

Summation Notation

A concise mathematical notation using the upper-case Greek letter sigma ($\sum$) to denote the sum of a sequence of terms.

Constant Multiple Rule for Summation

An algebraic property stating that a constant factor can be factored outside of a summation: $\sum_{r=m}^{n} c \times a_r = c \sum_{r=m}^{n} a_r$.

Sum of First $n$ Natural Numbers

The formula for the sum of consecutive integers from $1$ to $n$, given by $\sum_{r=1}^{n} r = 1 + 2 + 3 + \dots + n = \frac{n(n + 1)}{2}$.

Sum of First $n$ Odd Numbers

The identity stating that the sum of the first $n$ positive odd integers equals $n^2$: $1 + 3 + 5 + \dots + (2n - 1) = n^2$.

Sum of First $n$ Even Numbers

The identity stating that the sum of the first $n$ positive even integers equals $n^2 + n$: $2 + 4 + 6 + \dots + 2n = n^2 + n$.

Sum of Geometric Series

The closed-form formula for the sum of the first $n$ terms of a geometric sequence: $a + ar + ar^2 + \dots + ar^{n - 1} = \frac{a(1 - r^n)}{1 - r}$ where $r \neq 1$.

Factorial Series Sum

The identity for the sum of products of an index and its factorial: $\sum_{r=1}^{n} r(r!) = (1)(1!) + 2(2!) + \dots + n(n!) = (n + 1)! - 1$.

Divisibility Proof by Induction

A technique using mathematical induction to prove an expression $f(n)$ is divisible by an integer $d$ by writing $f(k + 1)$ as a multiple of $d$ plus a multiple of $f(k)$.

Bernoulli's Inequality

The fundamental inequality stating that for any real number $x > -1$ (or $x \ge 0$) and natural number $n \in \mathbb{N}$, $(1 + x)^n \ge 1 + nx$.

Sum of Squares of First $n$ Odd Integers

The sum formula for odd squares: $1^2 + 3^2 + \dots + (2n - 1)^2 = \frac{n}{3}(4n^2 - 1)$.

Sum of Squares of First $n$ Even Integers

The sum formula for even squares: $2^2 + 4^2 + \dots + (2n)^2 = \frac{2n}{3}(2n^2 + 3n + 1)$.

Combined Square Series Sum

The total sum of all consecutive squares from $1^2$ up to $(2n)^2$, computed by adding odd and even square sums: $\sum_{r=1}^{2n} r^2 = \frac{n}{3}(8n^2 + 6n + 1)$.