Finding the n-th Term and Sum of Sequences

Today in class, we learned how to find the nth term of an arithmetic sequence and also how to find the sum of a sequence.
I will start of by explaining the process for finding the nth term of an arithmetic sequence. To provide an example, I will work with the following sequence: 2, 4, 6, 8, ... n. To find the nth term of this sequence, we must use the following equation:

In this equation, a represents the first time of the sequence, and d represents the common difference. The common difference is "the constant added to each element of an arithmetic progression to obtain the next" (definition from: http://en.wiktionary.org/wiki/common_difference).

In this case, a is equal to 2, since 2 is the first term in our sequence. Also, d is equal to 2, because we must add 2 to each term in order to obtain the next term.

Therefore, our equation looks like this:

2 + (n - 1)2

(or, when simplified:)

2n

Now, if we would like to find the 10th term of this sequence, we simply substitute n with 10 in the equation, yielding:

2(10) = 20

Next, I will explain the process of finding the sum of n terms of a sequence (I will use the same sequence as above for this example also).

Taking a closer look at this equation, we can see that this is the same as the following:

(n/2)(a₁ + a_n) 

To obtain a_n, or the nth term, we can simply use the first equation.

Again, let us set n = 10. This means that a_n = 20.

All we have to do now is substitute these values into the sum equation:

((10)/2)(2 + 20) = 5(22) = 110

Now, we have determined the sum of this arithmetic sequence as well.