In the expansion of $(k + ax)^4$, where $a$ and $k$ are positive constants, the coefficient of $x$ is $128$ and the coefficient of $x^2$ is $24$. Find the values of $a$ and $k$.
$a = 0.5$ and $k = 4$
For the points $A(-5,-2))$, $B(3,1)$ and $C(-3, 4)$, - Find the equation of the line $AB$ giving your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers.
- Find the midpoint of $AB$.
- Calculate the length of $AC$ giving your answer as a simplified surd.
- Determine whether $AC$ is perpendicular to $BC$.
- $3x - 8y - 1 = 0$
- $(-1, -0.5)$
- $2\sqrt{10}$
- gradients are $3$ and $-0.5$ not perpendicular
Given $A = 80\mathrm{e}^{-0.02t}$, - Find $t$ when $A = 20$.
- Find the rate at which $A$ is decreasing when $t = 30$.
Prove that the sum of two consecutive odd numbers is always the difference between two square numbers.
Sum: $(2n-1) + (2n+1) = 4n$. Squares: $(n+1)^2 - (n-1)^2 = 4n$
Disprove the statement $3^n + 2$ is prime for all integers $n \geq 0$
$3^5 + 2 = 245$