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
, 10 =
, 15 =
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_1 t_1](https://s0.wp.com/latex.php?latex=t_1&bg=ffffff&fg=000000&s=0)
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
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
=
+ d(n-1)
- The nth-term is
= ![t_{50} t_{50}](https://s0.wp.com/latex.php?latex=t_%7B50%7D&bg=ffffff&fg=000000&s=0)
- d= -3
= 12
Now we insert all the known numbers into the formula
=
+ d(n-1)
= 12 + (-3)(50-1)
= 12 + (-3)(49)
= 12 -147
= -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
.
means
to
will be added.
The formula to determine the sum is ![S_n = \frac {n}{2}(t_1 + t_n) S_n = \frac {n}{2}(t_1 + t_n)](https://s0.wp.com/latex.php?latex=S_n+%3D+%5Cfrac+%7Bn%7D%7B2%7D%28t_1+%2B+t_n%29&bg=ffffff&fg=000000&s=0)
Insert into formula
![S_n = \frac {n}{2}(t_1 + t_n) S_n = \frac {n}{2}(t_1 + t_n)](https://s0.wp.com/latex.php?latex=S_n+%3D+%5Cfrac+%7Bn%7D%7B2%7D%28t_1+%2B+t_n%29&bg=ffffff&fg=000000&s=0)
![S_{50} = \frac {50}{2}( 12 + (-135)) S_{50} = \frac {50}{2}( 12 + (-135))](https://s0.wp.com/latex.php?latex=S_%7B50%7D+%3D+%5Cfrac+%7B50%7D%7B2%7D%28+12+%2B+%28-135%29%29&bg=ffffff&fg=000000&s=0)
![S_{50} = 25( -123) S_{50} = 25( -123)](https://s0.wp.com/latex.php?latex=S_%7B50%7D+%3D+25%28+-123%29&bg=ffffff&fg=000000&s=0)
![S_{50} = - 3,075 S_{50} = - 3,075](https://s0.wp.com/latex.php?latex=S_%7B50%7D+%3D+-+3%2C075&bg=ffffff&fg=000000&s=0)
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.