A recursive formula creates a sequence where each term is defined by the term(s) that precede it. In other words, in order to know term 12, you have to know term 11, etc.

\(\displaystyle a_n=(a_{n-1})^2; a_1=2\)

The problem already tells us that the first term is 2. Let's find the second term.

\(\displaystyle a_2=(a_{2-1})^2=(a_1)^2=2^2=4\)

We continue to find the rest of the terms in this way.

\(\displaystyle a_3=(a_{3-1})^2=(a_2)^2=4^2=16\)

\(\displaystyle a_4=(a_{4-1})^2=(a_3)^2=16^2=256\)

Apply the rule to find the first few terms:

\(\displaystyle a_{1} = 1\)

\(\displaystyle a_{2} = -1\)

\(\displaystyle a_{3} = a_{2} a_{1} = -1 \cdot 1 = -1\)

\(\displaystyle a_{4} = a_{3} a_{2} = -1 \cdot \left (-1 \right ) = 1\)

\(\displaystyle a_{5} = a_{4} a_{3} = 1 \cdot \left (-1 \right ) = - 1\)

\(\displaystyle a_{6} = a_{5} a_{4} = -1 \cdot 1 = - 1\)

After the sixth term, it is apparent that this cycle will repeat itself, so the first twenty terms of the sequence will be, in order:

\(\displaystyle \left \{ 1, -1, -1, 1, -1, -1,1, -1, -1, 1, -1, -1, 1, -1, -1,1, -1, -1,1, -1\right \}\)

Seven of these first twenty terms are positive.

Apply the rule to find the first few terms:

\(\displaystyle a_{1} = -5\)

\(\displaystyle a_{2} = 2 a_{ 1} +6 = 2 (-5) + 6 = -10 + 6 = -4\)

\(\displaystyle a_{3} = 2 a_{ 2} +6 = 2 (-4) + 6 = -8 + 6 = -2\)

\(\displaystyle a_{4} = 2 a_{ 3} +6 = 2 (-2) + 6 = -4 + 6 = 2\)

The first positive term of the sequence is \(\displaystyle a_{4}\).

In order to determine which expression describes the sequence in the problem, we must determine the relationship each entry has with its position in the sequence. For example, for n=1, we must determine which expression involving n will yield a result of 3, for n=2, we must determine which expression will yield a result of 8, and so on, ensuring that the expression holds true for every n value in the sequence. If we check each of our answers, we can see that only the following expression will give the correct result for each increasing value of n:

\(\displaystyle n^2+2n\)

\(\displaystyle (1)^2+2(1)=3\)

\(\displaystyle (2)^2+2(2)=8\)

\(\displaystyle (3)^2+2(3)=15\)

If we continue, we can see that we will obtain the sequence 3,8,15,24,35,48,63,..., so this expression is the correct representation of the sequence given in the problem.

To find the mean of a set of numbers we first must add all the numbers together.

\(\displaystyle 99 + 70 + 84 + 63 + 91 + 59 = 466\)

Using the formula for mean we get,

\(\displaystyle \text{Mean}=\frac{\text{Sum of Numbers}}{\text{Total Number of Entries}}\)

Therefore we get,

\(\displaystyle \text{Mean}=\frac{446}{6}=77.66=77.7\)

The median of a set of numbers is the middle value of the set.

To find the middle value of this particular data set put the prices in order of lowest to highest

\(\displaystyle 36, 52, 57, 83\)

Since we have an even number of entries we will need to find the mean of the two middle numbers and this will become our median.

\(\displaystyle 52 + 57 = 109\)

\(\displaystyle 109 / 2 = 54.5\)

The sequence goes down by 2 so,

 \(\displaystyle 7 - 2 = 5 - 2 = 3 - 2 = 1 - 2 = -1\).

Therefore the next number in the sequence is \(\displaystyle -1\).

The sequence goes up by 5 so,

 \(\displaystyle 15 + 5 = 20 + 5 = 25 + 5 = 30 + 5 = 35 + 5 = 40\)

Therefore the next term in the sequence will be \(\displaystyle 40\).

To find the value related to the specific percentage we need to set up a proportion and solve for x.

\(\displaystyle \frac{15}{100}=\frac{x}{60}\)

From here we cross multiply and divide to find the value of x.

\(\displaystyle 15 \times 60 = 900\)

\(\displaystyle \frac{900}{100}=9\)

To find the value for a specific percentage of a number we first need to convert the percentage into a decimal.

\(\displaystyle 50/100 = 0.5\)

From here we multiply the decimal with the number we are given in the question.

\(\displaystyle 0.5 \times 135 = 67.5 = 68\)