Week 1 Arithmetic Sequences

This week was the first week of Pre Calculus 11 for me. So far it has been good, as we are learning about series and sequences. Today I am going to teach you the difference between arithmetic sequences and an arithmetic series. I will also show the formula to find the nth-term and how to find the sum of the terms. STAY TUNED 🙂

A SEQUENCE for example, is a set of numbers that are changing in some way.

ex/ 5, 10, 20, 40, 80

5 is called $t_1$ , 10 = $t_2$ , 15 = $t_3$ and so on.    The t stands for term.

An ARITHMETIC SEQUENCE is a sequence that changes by a constant amount. The constant amount is also known as the common difference.

ex/ 5, 10, 15, 20, 25,     d=+5 ;The common difference is +5

to find the common difference the formula is  $t_2$  –  $t_1$

NOT ARITHMETIC,       NO CONSTANT DIFFERENCE

An ARITHMETIC SERIES is the terms of an arithmetic sequence added together. The point is to find the sum of the desired terms.

ex/ 5+10+15+20+25 = 75

How to find the nth-term

Say you wanted to find what $t_{50}$ is in the sequence 12, 9, 6, 3, 0 is but you don’t want to continue writing the whole sequence out. Well your in luck, there is a fast way.

The formula to find the nth-term is  $t_n$ = $t_1$ + d(n-1)

• The nth-term is $t_n$$t_{50}$
• d= -3
• $t_1$ = 12

Now we insert all the known numbers into the formula

1. $t_n$ = $t_1$ + d(n-1)
2. $t_{50}$ = 12 + (-3)(50-1)
3. $t_{50}$ = 12 + (-3)(49)
4. $t_{50}$ = 12 -147
5. $t_{50}$ = -135

There you are, you just figure out how to find the 50th term quick and easy.

How to find the sum of a series

We will take the example from above and determine the sum of  $S_{50}$$S_{50}$ means $t_1$ to $t_{50}$ will be added.

The formula to determine the sum is $S_n = \frac {n}{2}(t_1 + t_n)$

• $S_n$$S_{50}$
• n = 50
• $t_1$ = 12
• $t_n$ = -135

Insert into formula

1. $S_n = \frac {n}{2}(t_1 + t_n)$
2. $S_{50} = \frac {50}{2}( 12 + (-135))$
3. $S_{50} = 25( -123)$
4. $S_{50} = - 3,075$

There you have it. Today you learned what an arithmetic sequence and series were, what the formula to find the nth-term and the sum of a series was, and how to solve. I hope this blog post helps you in your future with practice and studying.