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.

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

The sequence follows the pattern for the equation:

\(\displaystyle n_x=2n_{x-1}+3\)

\(\displaystyle n_{x-1}=77\)

Therefore,

\(\displaystyle n_x=2(77)+3=154+3=157\)

The first term for the sequence is where \(\displaystyle n=1\). Thus,

\(\displaystyle x_1=1+3(1)^2=4\)

So the first term is 4.  Repeat the same thing for the second \(\displaystyle \left ( n=2 \right )\), third \(\displaystyle \left ( n=3 \right )\), fourth \(\displaystyle \left ( n=4 \right )\), and fifth \(\displaystyle \left ( n=5 \right )\) terms.

\(\displaystyle x_2=1+3(2)^2=13\)

\(\displaystyle x_3=28\)

\(\displaystyle x_4=49\)

\(\displaystyle x_5=76\)

We see that the first five terms in the sequence are

\(\displaystyle 4,13,28,49,76\)

To go from \(\displaystyle a_{n-1}\) to \(\displaystyle a_{n}\), we're adding 1 to the denominator. In words, we're flipping \(\displaystyle a_{n-1}\), adding 1, then flipping it again. For example, to get from \(\displaystyle a_{4}=\frac{1}{4}\) to \(\displaystyle a_{5 } = \frac{1}{5}\) we would have to flip \(\displaystyle \frac{1}{4}\) to be 4, add 1 to get 5, then flip again to get \(\displaystyle \frac{1}{5}\).

The formula that shows this is

\(\displaystyle a_{n} =\frac{1}{\frac{1}{a_{n-1}}+1}\)

The problem isn't asking us to add the first 1,000 square numbers, but all the square numbers from 1 to 1,000. To figure out this sum, you might need to look at a list of square numbers, or play around with large squares to find the largest one under 1,000. This ends up being 31: \(\displaystyle 31^2 = 961\) while \(\displaystyle 32^2 = 1024\), which is not between 1 and 1,000. So what we're adding is the first 31 square numbers.

This means we can plug 31 in for n in that formula:

\(\displaystyle \frac{31*(31+1)*(31*2 + 1 )}{6} = \frac{31*32*63}{6 } = 10,416\)

To find the sum of all the integers in between 17 and 8,043, first we will find the sum of every integer from 8,043, and then we will subtract out the sum of the numbers 1-16, since those aren't between 17 and 8,043.

The sum of the first 8,043 integers is \(\displaystyle \frac{(8043)(8044)}{2 } = 32, 348, 946\)

The sum of the integers 1-16 is \(\displaystyle \frac{(16)(17)}{2 } = 136\)

Subtracting gives us \(\displaystyle 32, 348, 810\)

To calculate this sum, first we will need to find the sum of the positive integers, then the negative interers, then add them together.

To find the sum of the positive integers, use the formula with \(\displaystyle n = 4,400\):

\(\displaystyle \frac{(4400)(4401)}{2 } = 9,682, 200\)

To find the sum of the negative integers, we can use the same formula as the positive numbers and then just make that answer negative.

\(\displaystyle \frac{(2256)(2257)}{2 } = 2,545, 896\) so the negative numbers add up to \(\displaystyle -2,545,896\).

The final answer is \(\displaystyle 9,682, 200 - 2,545, 896 = 7,136, 304\)