Math 11

Week 1 Arithmetic Sequences

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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.

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