MaxWong

Determine whether each number is rational or not, so you can determine its function values.

f(-8) + f(-pi) + f(50) + f(9/2) = -8 + pi^2 + 50 + 9/2 = 93/2 + pi^2

You have d = 1/2, and the first equation just says a = (5 - 8d)/2. So you can substitute your answer into the first equation to get a.

In general cases, you can treat complex numbers as vectors because there is a one-to-one correspondence \(\phi:\mathbb C \to \mathbb R^2\) defined by \(\phi (z) = \begin{bmatrix}\operatorname{Re}(z)\\\operatorname{Im}(z)\end{bmatrix}\).

You just have to be careful that if \(BC = 2AB\), that actually means \(\left\|\overrightarrow{BC}\right\| = 2 \left\|\overrightarrow{AB}\right\|\), not \(\overrightarrow{BC} =2 \overrightarrow{AB}\). This goes for any equation with lengths.

You treated BC and AB as vectors. If \(\overrightarrow{BC} = 2\overrightarrow{AB}\), that means \(\overrightarrow{BC}\) is parallel to \(\overrightarrow{AB}\), which is not what you want, since \(\angle ABC = 90^\circ\).

Using 2i instead of 2 gives the vectors a rotation of 90 degrees, which gives the correct answer.

The multiplication by -2i gives a rotation of 90 degrees clockwise, maybe the resulting point C is not in the first quadrant, which isn't what the question wanted.

And what do you intend to do with that function?

What did you get after simplifying the equation?

Since a = (5 - 8d)/2, substituting this into the second equation will completely eliminate all "a"s in the second equation, giving \(\left(\dfrac{5 - 8d}2 + 3d\right)\left(\dfrac{5 - 8d}2+4d\right) = 5\)

Expand and simplify, and you will get an equation entirely in terms of d. Now you can solve it to get the value of d and a.

Once you get the value of d from that equation above, the value of a is just (5 - 8d)/2.

The function is not injective, so its inverse function does not make sense.

Please use a calculator.

\(5\times 5 \times 76 = 1900\)

Let a be the first term and d be the common difference of the arithmetic sequence.

\(\begin{cases} (a + d) + (a + 7d) = 5\\ (a + 3d)(a + 4d) = 5 \end{cases}\)

\(\begin{cases} a = \dfrac{5 - 8d}{2}\\ (a + 3d)(a + 4d) = 5 \end{cases}\)

Now you can substitute the first equation into the second, and solve the resulting quadratic equation for the value(s) of d.

Then you will also be able to find the corresponding value(s) of a.

The answer is a + 19d, but please calculate the value(s) of a and d and substitute to get the numerical answer.