Begin by considering how much the numbers change between each set of values.  The following places the change in brackets:

$\displaystyle 4,[+10]14,[+15]29,[+20]49,...$

You can guess that the next change will be by $\displaystyle 25$.  Therefore, the next value in the sequence will be $\displaystyle 49+25$ or $\displaystyle 74$.

Start by filling in the changes that happen from number to number in the sequence.  This is done below in brackets:

$\displaystyle 17,[+4]21,[+6]27,[-2]25,[+4]29,[+6]35,?,37$

Now, you know that $\displaystyle 35-2+4=37$

Thus, you know that the sequence will continue after $\displaystyle 35$ by subtracting $\displaystyle 2$.  Thus, the missing value is $\displaystyle 33$.

Notice that the numbers are increasing by 6 every time.  

Add 6 to the end of the last number to obtain the number in the question mark.

$\displaystyle -2+6 = 4$

The answer is:  $\displaystyle 4$

Recall that in an arithmetic sequence, the difference between terms is constant. From the given first and second terms, we see that the sequence is increasing by $\displaystyle 26$ for each term. Thus, the third term must be $\displaystyle 177$, and the fourth term must be $\displaystyle 203$.

To generate the sequence, begin with 4, then alternately multiply by 3 and add 2:

$\displaystyle \textbf{4 }\times 3 = \textbf{12 }$

$\displaystyle 12+ 2 = \textbf{14}$

$\displaystyle 14 \times 3 = \textbf{42}$

$\displaystyle 42 + 2 = \textbf{44}$

$\displaystyle 44 \times 3 = \textbf{132 }$

The next entry:

$\displaystyle 132 + 2 = \textbf{134 }$,

the correct choice.

Find the missing number in the sequence:

$\displaystyle 7,..., 343, 2401,16807$

We have a sequence where each term is a multiple of the previous term. This is known as a geometric series. 

We need to find the common multiple, and then use it to find our second term. To find the common multiple, divide any term by its previous term.

ex) 

$\displaystyle \frac{2401}{343}=7$

So, our common multiple is 7. Use this to find our second term.

$\displaystyle 7*7=49$

So, our answer is 49

Find the missing term in the following arithmetic series:

$\displaystyle 18,31,...,57,70,83$

An arithmetic series is one in which the next term is found by adding a constant number to the previous term. This is called the common difference.

First, we need to find the common difference. Do so by subtracting any term from its following term.

$\displaystyle 83-70=13$

Note that it doesn't matter which pair of consecutive terms we choose, just so long as we subtract the smaller from the larger.

So, our common difference is 13. Thus, our answer can be found via the following:

$\displaystyle 31+13=44$

So our answer is 44

Recall that in an arithmetic sequence, we will be adding or subtracting the same number to get each successive term.

We can tell that the terms are decreasing by $\displaystyle 3$ each time. Thus, we can make a table to figure out the tenth term.

$\displaystyle \begin{tabular}{|c|c|} \hline\text{Term Number}&\text{Value}\\ \hline 1 &11\\ 2 &8\\ 3 &5\\ 4 &2\\ 5 &-1\\ 6 &-4\\ 7 &-7\\ 8 &-10\\ 9 &-13\\ 10 &-16\\ \hline \end{tabular}$

The tenth term is $\displaystyle -16$.

Recall that in an arithmetic sequence, you will be adding or subtracting by a certain number to get the subsequent values in the sequence.

From the given numbers, you should notice that the sequence is increasing by $\displaystyle 7$ every time. Thus, the fourth number in the sequence should be $\displaystyle 26$. The fifth number in the sequence must be $\displaystyle 33$.

The pattern of the sequence is $\displaystyle \small x_n=-2x_{n-1}+1$.

Therefore,

$\displaystyle x_{n}=-2(-21)+1=43$