← Here is equal to plus because is the sum of the pervious term. So if I add which is term number 1, it will give me the . I will use this formula to find out the sum’s up to n = 10.
← Now, we can sub as 1 and as
Here n = 2, so
I did this for where
On the other hand, I used Microsoft Excel 2010 to do it for me too. I got the same results. The method to do this is shown in the appendix.
Because I got the same result, I will use Excel to graph. The method is shown in the Appendix.
By looking at this graph, I can say that when value is past 10, the is 2 and remains as it is. So, as the value increases, the remains the same.
Now I will examine another sequence, and I will find out the sum of , where , , and where it goes from 0 to 10 . So it should look like this:
I will find out the for .
In order to find , I will only use Microsoft Excel 2010 since I know that it gave me the right result before. I will plug in the equation , where the is 1, is 3 and value is from 0 to 10. The method is shown in the appendix.
To find the results, I just subbed in 3 instead if . Now I will graph it using the same process:
By looking at this graph, I can say that when value is past 10, is 3 and it remains constant. This suggests that when approaches (infinity), sum of the infinity value is 3.
After looking at the two sum, when, and when , I think that the general term is
Where is the sum of the infinite numbers. I came up with this formula because we know,

For , when kept increasing, the sum of the value remained the same and that’s why . We can say that instead of because can go up to and on the other hand is equivalent to 2.

For , when kept increasing, the sum of the value remained the same and that’s why . We can say that instead of because can go up to and on the other hand is equivalent to 3.
Thus,
In order to test this general statement, I am going to let be 4. So if is 4, then . So now I am going to use Microsoft Excel 2010 again to prove this general statement. The method is shown in the appendix.
Now I will graph it so that it is easier to see and check:
As we can see that the graph is increasing but as value passes 10, we see that remains 4. This proves my general term as I predicted that
Now I will sub different values for both and , in the equation
Here, I will let be the sum of the first terms for various values of and For example, be the sum of the first 9 terms when and .
But before we do anything, I am going to change to make it look simpler –
I am not sure if this is going to work. So I am going to test it using my TI84 Plus
Let
Let
Let
Let
Let
Let
This proves that
After looking at this equation, I think the general term will be . Where is the sum of the infinite numbers when approaches . The reason behind this is that, we figured when where . But now, we are using . For example, let and . This is what it should look like –
Here we can see that , so therefore the result should be
But since we have define as the sum, we should rewrite it as .
But I have only predicted this equation, in order to test it, I will use Microsoft Excel 2010. The process is shown in the appendix.
is the sum of the first term, for various values of and .
is the sum of the first term, when and .
Here, is the term number or the term, is 2 and is 2 from my equation, is the equation subbing different values of , in this case is is defined as . And is defined as the sum of the first term. In this case, , meaning the value will keep increasing, but the will increase up to 4 unit, and then it will remain constant. It won’t go further than 4. I will use a graph and this is explain more –
Now, I am going to predict that when and in the equation , the .
So according to my guess, I was totally right. This proves that the general term is .
So now I am going to use my calculator to figure that what happens when
So, I am going to let , and in .
I will use my TI84 Plus in order to find out the situation. It look like this–
After I pressed ENTER key, I got this result –
This suggest there is no result when . The reason behind this is that, when we punch in or , we get the same error. As we cannot solve it.
Now I am going to let a, and in the equation .
I will use my TI84 Plus in order to find out the situation. It looks like this –
So this means, when I use as a negative number it will show some results. I will check if the result shown above different from the equation .
This means, that when it is equal to when . But what happens when is a negative number and is a odd number. For example, when .
I will let , and in the equation .
I will again use my TI84 Plus to find out the result –
This is what I get when I hit enter –
This simple means that you cannot solve a equation when there is a negative number. For example,
In conclusion, when is a negative number, had to be an even number to change the negative sign to positive sign.
In short, , , can be both positive and negative number, can also be both negative and positive number but only when is an even number.