Step 1: We know it's a geometric sequence, so the difference between each consecutive term is the same. We must find the difference.



Step 2: For every term after the first term, we will add a multiple of  to the first term. 

Step 3: We can write an equation that helps us find the nth term of the sequence. We add a multiple of 4n to 5; the equation becomes .

Step 4: We need to find the value of n. We want the 15th number and we are given the first number. There are  numbers in between, so n=14.

Step 5: Plug in 14 for n and find the 15th term



The  term is 

Step 1: We need to find the difference between the terms. To find the difference, subtract the first two terms.

Difference=

Step 2: To find the next term of a arithmetic sequence, we add the difference to the previous term.

Fourth term= term+5
 term=. The fourth term is 19.

Step 3: To find the  term, we must add a multiple of  to the first term. In this problem, we are given the  term, so we need to find how many terms are in between the  and  term.

. This tells me that I have to add  fourteen times to get to the  term. .

Step 4: Now that we know what the value of n, we can plug it in to the equation:

, where 4 is the starting number, and n represents how many times I need to add 5 to the next term (and up to the 15th term).

Let's plug it in:

.

So, the  term in this sequence is .

Step 1: Try to find a pattern in the sequence

The pattern is decreasing perfect squares, the difference between each base of the consecutive term is ..

Step 2: Find the next term...

If the base of the next term goes down by , the next term is ..



The next term of this sequence is .

Step 1: Try to find the difference between the terms.

Difference between  and  is .
Difference between  and  is .
Difference between  and  is .

Difference between  and  is .

The difference between consecutive numbers is an odd number, and the difference between the consecutive terms is always ..

The difference between  and  is 



The next number in the sequence is .

Step 1: Find the next term in the sequence: 

Step 2: Can you recognize the sequence here??

This is the Fibonacci Sequence.. The sum of the previous two terms is equal to the next consecutive term..







The missing term is 13.

Step 1: Find the difference between each two consecutive terms...

The difference is always 6 because we have a sequence...

Step 2: Find the next term in the sequence..



The next term is .

Step 1: Calculate the difference between the terms...

, add 9
, add 9
, add 9

Step 2: We found the pattern between the terms..

So, we can add 9 to the term before the  to find the value of ..



The missing term (the next term) in the sequence is .

Step 1: Find the difference between each term...

Subtract the first term from the third term...


There are two terms between first and third...Take the answer in step 1 and divide by 2 to get the difference between consecutive terms...



The common difference is .

Step 2: Find an equation that describes the sequence....

The equation is , where  represents how many terms I need to calculate and  is the first term... 

Step 3: Plug in ...

To find n, we subtract the term that we want from the original term...

So, if we want the th term and we are given the first term...

Then 

So, 

The th term is .

Step 1: Find the successive difference rows until we get equal values between every two consecutive numbers.

Step 2: Since we obtain equal values for the successive differences in the second row, hence the  term of the sequence is a second-degree polynomial. 

So, the  term takes the form of

Step 3: Now, substituting  into the formula, we get:

Step 4: Solve the system of linear equations:

So, the   term takes the form:

Step 5: Plug in  into the form...

The ratio test fails when . Otherwise the series converges absolutely if , and diverges if .

Testing the series , we have





Hence the ratio test fails here. (It is likely obvious to the reader that this series diverges already. However, we must remember that all intuition in mathematics requires rigorous justification. We are attempting that here.)