Sequencing problems require us to look at the numbers given to us ad decipher a pattern.

7 – 4 = 3 so 3 was added to the first number (4)

11 – 7 = 4 so 4 was added to the second number (7)

16 – 11 = 5 so 5 was added to the third number (11)

If it is to continue in this pattern, then 6 should be added to 16, yielding 22 as the correct answer

A sequence is simply a list of numbers that follow some sort of consistent rule in getting from one number in the list to the next one.  Sequences generally fall into three categories:  arithmetic, geometric, or neither.

In arithmetic sequences, I add the same number each time to get from one number to the next.  In other words, the difference between any two consecutive numbers in my list is the same.

In geometric sequences, I multiply by the same number each time to get from one number to the next.  In other words, the ratio between any two consecutive numbers in my list is the same.

Finally, sequences that are neither, still follow some rule, but it just happens not to be one of these two.

Looking at our sequence, we might quickly notice that each number is simply 7 more than the number before.  In other words, I can find the next number by adding 7 each time.  Hence, our sequence is arithmetic.

Unfortunately, we need to find the 50th term in this sequence, and the problem only got us through the first four.  A simple (yet way too time-consuming approach) would be to keep adding 7 until we get to term number 50.  Not only is that the long way, we also risk losing count and ending up on the wrong term.  So what's the easier way?

The easier way hinges on the fact that I am simply adding 7 over and over again.  If I want to find the 2nd term, I start with the 1st term and add 7 once.

\(\displaystyle -8+7=-1\)

To find the 3rd term, I add 7 twice.

\(\displaystyle -8+7+7=6\)

You might already see the pattern.  For the 4th term I would add 7 three times, for the 5th four times, 6th five times, etc.

Notice that to find any term, I simply add 7 one less time than the number of the term.  Therefore, to find the 50th term, I would add 7 forty-nine times.

But adding 7 forty-nine times is the same as adding forty-nine 7s.  But forty-nine 7s are the same as 49 times 7.

\(\displaystyle 49\cdot7=343\)

Therefore, to find the 50th term, I simply need to add 343 to our starting value.

\(\displaystyle -8+343=335\)

Because it is an arithmetic sequence, the difference between each term is the same. Therefore find out the difference between any two consecutive terms. \(\displaystyle 12-7 = 5\) and so the sequence increases by 5 each term. Thus the answer is \(\displaystyle 12 + 5 = 17\).

We can represent four consecutive integers using the following expressions:

 \(\displaystyle x,\ x+1,\ x+2, \textup{ and }x+3\)

Create an equation using the information in the problem.

 \(\displaystyle x + (x+1) + (x+2) + (x+3) =42\)

\(\displaystyle 4x + 6 = 42\)

Subtract 6 from both sides of the equation.

\(\displaystyle 4x+6-6=42-6\)

\(\displaystyle 4x=36\)

Divide both sides of the equation by 4.

\(\displaystyle \frac{4x}{4}=\frac{36}{4}\)

\(\displaystyle x=9\)

The highest number in the series is the x-variable plus 3. We can write the following:

\(\displaystyle x+3 = 12\).

To get the next term in the sequence you subtract a decreasing amount from the preceding term. You subtract 7 from 40 to get 33, then 6 from 33 to get 27, and so on until you subtract 2 from 15 to get 13.

The 295th day would be the day after the 42nd week has completed. 294 days/7 days a week = 42 weeks. The next day would therefore be a monday.

The sequence is multiplied by \(\displaystyle \small 2\pi\) each time.

The formula for finding the \(\displaystyle n^{th}\) term of an arithmetic sequence is as follows:

\(\displaystyle a_n=a_1+d(n-1)\)

where

\(\displaystyle d\)= the difference between consecutive terms

\(\displaystyle n\)= the number of terms

Therefore, to find the \(\displaystyle 10th\) term:

\(\displaystyle a_{10}=-3+(-12)(10-1)\)

\(\displaystyle a_{10}=-3+(-12)(9)\)

\(\displaystyle a_{10}=-3-108\)

\(\displaystyle a_{10}=-111\)

Notice that between each of these numbers, there is a difference of \(\displaystyle 7\); however the first number is \(\displaystyle 10 + 7\), the second \(\displaystyle 10 + 7 * 2\), and so forth. This means that for each element, you know that the value must be \(\displaystyle 10 + 7*n\), where \(\displaystyle n\) is that number's place in the sequence. Thus, for the \(\displaystyle 93\)rd element, you know that the value will be \(\displaystyle 10 + 7*93 = 10 + 651\) or \(\displaystyle 661\).

Notice that between each of these numbers, there is a difference of \(\displaystyle 6\). This means that for each element, you will add \(\displaystyle 6\). The first element is \(\displaystyle 6-2\) or \(\displaystyle 4\). The second is \(\displaystyle 6*2-2\) or \(\displaystyle 10\), and so forth... Therefore, for the \(\displaystyle 20\)th element, the value will be \(\displaystyle 6*20-2\) or \(\displaystyle 118\).