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

$\displaystyle Difference=9-5=4$

Step 2: For every term after the first term, we will add a multiple of $\displaystyle 4n$ 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 $\displaystyle 4n+5$.

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

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

$\displaystyle 15th\ Term=5+4(14)=5+56=61$

The $\displaystyle 15th$ term is $\displaystyle 61$

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

Difference=$\displaystyle 9-4=5$

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

Fourth term=$\displaystyle 3rd$ term+5
$\displaystyle 4th$ term=$\displaystyle 14+5=19$. The fourth term is 19.

Step 3: To find the $\displaystyle nth$ term, we must add a multiple of $\displaystyle 5n$ to the first term. In this problem, we are given the $\displaystyle 1st$ term, so we need to find how many terms are in between the $\displaystyle 15th$ and $\displaystyle 1st$ term.

$\displaystyle 15-1=14$. This tells me that I have to add $\displaystyle 5$ fourteen times to get to the $\displaystyle 15th$ term. $\displaystyle n=14$.

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

$\displaystyle 4+5n$, 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:

$\displaystyle 4+5(14)=4+70=74$.

So, the $\displaystyle 15th$ term in this sequence is $\displaystyle 74$.

Step 1: Try to find a pattern in the sequence

$\displaystyle 196=14^2$
$\displaystyle 144=12^2$

$\displaystyle 100=10^2$
$\displaystyle 64=8^2$

The pattern is decreasing perfect squares, the difference between each base of the consecutive term is $\displaystyle 2$..

Step 2: Find the next term...

If the base of the next term goes down by $\displaystyle 2$, the next term is $\displaystyle 6^2$..

$\displaystyle 6^2=36$

The next term of this sequence is $\displaystyle 36$.

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

Difference between $\displaystyle 0$ and $\displaystyle 3$ is $\displaystyle 3$.
Difference between $\displaystyle 3$ and $\displaystyle 8$ is $\displaystyle 5$.
Difference between $\displaystyle 8$ and $\displaystyle 15$ is $\displaystyle 7$.

Difference between $\displaystyle 15$ and $\displaystyle 24$ is $\displaystyle 9$.

The difference between consecutive numbers is an odd number, and the difference between the consecutive terms is always $\displaystyle 2$..

The difference between $\displaystyle ?$ and $\displaystyle 24$ is $\displaystyle 9+2=11$

$\displaystyle ?=24+11=35$

The next number in the sequence is $\displaystyle 35$.

Step 1: Find the next term in the sequence: $\displaystyle 1,1,2,3,5,8,?$

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..

$\displaystyle 1+1=2$
$\displaystyle 1+2=3$
$\displaystyle 2+3=5$
$\displaystyle 3+5=8$
$\displaystyle 5+8=?, ?=13$

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..

$\displaystyle 27+6=33$

The next term is $\displaystyle 33$.

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

$\displaystyle 7\rightarrow16$, add 9
$\displaystyle 16\rightarrow25$, add 9
$\displaystyle 25\rightarrow34$, add 9

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

So, we can add 9 to the term before the $\displaystyle x$ to find the value of $\displaystyle x$..

$\displaystyle 43+9=52$

The missing term (the next term) in the sequence is $\displaystyle 52$.

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

Subtract the first term from the third term...

$\displaystyle \large 8-2=6$
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...

$\displaystyle \large \frac {6}{2}=3$

The common difference is $\displaystyle \large 3$.

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

The equation is $\displaystyle \large 3n+2$, where $\displaystyle \large n$ represents how many terms I need to calculate and $\displaystyle \large 2$ is the first term... 

Step 3: Plug in $\displaystyle \large n$...

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

So, if we want the $\displaystyle \large 16$th term and we are given the first term...

Then $\displaystyle \large n=16-1=15$

So, $\displaystyle \large 3(15)+2=45+2=47$

The $\displaystyle \large 16$th term is $\displaystyle \large 47$.

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

$\displaystyle 4,11,20,31$

$\displaystyle 7,9,11$

$\displaystyle 2,2$

Step 2: Since we obtain equal values for the successive differences in the second row, hence the $\displaystyle n^{th}$ term of the sequence is a second-degree polynomial. 

So, the $\displaystyle n^{th}$ term takes the form of

$\displaystyle a_n=An^2+Bn+C$

Step 3: Now, substituting $\displaystyle n=1,2,3$ into the formula, we get:

$\displaystyle A+B+C=4...........(1)$

$\displaystyle 4A+2B+C=11........(2)$

$\displaystyle 9A+3B+C=20.........(3)$

$\displaystyle (2)-(1):3A+B=7.......(4)$

$\displaystyle (3)-(2):5A+B=9.......(5)$

$\displaystyle (5)-(4):2A=2$

$\displaystyle A=1$

$\displaystyle B=7-3(1)=4$

$\displaystyle C=4-4-1=-1$

Step 4: Solve the system of linear equations:

$\displaystyle A=1, B=4, C=-1$

So, the  $\displaystyle n^{th}$ term takes the form:

$\displaystyle n^2+4n-1$

Step 5: Plug in $\displaystyle n=32$ into the form...

$\displaystyle (32)^2+4(32)-1$

$\displaystyle 1024+128-1$

$\displaystyle 1151$

The ratio test fails when $\displaystyle \lim_{k\to \infty} \left |\frac{a_{k+1}}{a_k} \right | =1$. Otherwise the series converges absolutely if $\displaystyle \lim_{k\to \infty} \left |\frac{a_{k+1}}{a_k} \right | < 1$, and diverges if $\displaystyle \lim_{k\to \infty} \left |\frac{a_{k+1}}{a_k} \right | >1$.

Testing the series $\displaystyle \sum_{k=0}^{\infty} k$, we have



$\displaystyle \lim_{k\to\infty} \left |\frac{a_{k+1}}{a_k} \right | = \lim_{k\to\infty} \left |\frac{(k+1)}{(k)} \right | = \lim_{k\to\infty} 1+ \frac{1}{k} = 1.$

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.)