The sum of the first five terms of an arithmetic progression is 5. The sum of the next ten terms is 175. Find the sum of the first twenty terms.

The sum of the first $n$ terms of an AP is given by $S_n = \frac{n}{2}(2a + (n-1)d)$, where $a$ is the first term and $d$ is the common difference.

Given: The sum of the first five terms ($S_5$) is 5.

Divide both sides by $\frac{5}{2}$:

Divide by 2:

Given: The sum of the next 10 terms is 175. This means the sum from the 6th term to the 15th term.
This can be written as $S_{15} - S_5 = 175$.
So, $S_{15} = S_5 + 175 = 5 + 175 = 180$.
Now use the formula for $S_{15}$:

Divide both sides by $\frac{15}{2}$:

Divide by 2:

We have two equations:

1. $a + 2d = 1$
2. $a + 7d = 12$

Subtract equation (1) from equation (2):

Substitute $d = \frac{11}{5}$ into equation (1):

Using the formula $S_n = \frac{n}{2}(2a + (n-1)d)$ with $n=20$, $a=-\frac{17}{5}$, and $d=\frac{11}{5}$:

The sum of the first twenty terms is $\boxed{350}$.

10. b) The gradient of a curve at the point $(x,y)$ is given by $ax^2 - 4x - 4$. If the minimum point on the curve is $(2,16)$, find (i) the value of $a$; (ii) the equation of the curve.

b)(i) the value of $a$;

At a minimum point, the gradient of the curve is zero.
Given the gradient $\frac{dy}{dx} = ax^2 - 4x - 4$.
The minimum point is $(2,16)$, so at $x=2$, $\frac{dy}{dx} = 0$.

The value of $a$ is $\boxed{3}$.

b)(ii) the equation of the curve.

Now we know $a=3$, so the gradient function is:

Substitute $x=2$ and $y=16$ into the equation of the curve:

Substitute $C=24$ back into the equation:

The equation of the curve is $\boxed{y = x^3 - 2x^2 - 4x + 24}$.