Week 1 – My Arithmetic Sequence

Part 1

This week, I have learned how to solve arithmetic sequence and series. Before we begin, we must learn some vocabulary to be able to solve these questions.

Arithmetic sequence: is a sequence of number that the difference between the consecutive terms is constant. Ex: The sequence 1 ,3, 5, 7, 9, …  is an arithmetic progression with common difference of 2.

General term: To determine an expression for the general term, we use t_n. To find what is  t_n, we use this formula:

t_n=t_1=d(n-2)

 

If we have a sequence going, -3, 2, 7, 12…

-3 = t_1, 2= t_2$, 7 = t_3 and so on.

The letter “n” is the number of term is needed to get to the number you are looking for. The letter “d” is the common difference between numbers. To determine “d” you can use this formula:

t_1+d=t_2

Now to begin, I’ll go through the whole formula step by step.

If we have the sequence of

10, 20, 30, 40, 50… ,

and the question is asking for the 50th term, we will begin by finding the common difference.

Common difference (d) = +10 each time.

Next, we would put everything in the formula.

t_n=t_1+d(n-1)

 

t_{50}=10+10(50-1)

 

t_{50}=10+10(49)

 

t_{50}=10+490

 

t_{50}=500

 

As a result, t_{50} is 500.

 

 

Part 2

Second part of this blog is how to find the series. Again lets go through some vocabulary.

Series: Is the sum of the terms in an arithmetic sequence.

Ex:

The arithmetic sequence 5, 8, 11, 14

The arithmetic series 5+8+11+14

The termS_n is to represent the sum of the first “n” terms of a series.The nth term of an arithmetic series is the nth term of the related arithmetic sequence.

S_nS_n

S_1S_1

S_2t_1t_2

S_3t_1t_2 t_3

 

Now, we can begin to find the series of my sequence

10, 20, 30, 40, 50,

However, we don’t know the formula to find S_n.

Formula to find S_n.

S_n=\frac{n(t_1+t_n)}{2}

Lets solve this problem. We will try to find the 50th term.

 

S_n=\frac{n(t_1+t_n)}{2}

 

S_{50}=\frac{50(10+t_500)}{2}

 

S_{50}=\frac{50(510)}{2}

 

S_{50}=\frac{25,500)}{2}

 

S_{50}=12,750

 

In conclusion, you have learned to find the arithmetic sequence and the arithmetic series. Stay tune for next week as I learn different a math during class.

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