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Arithmetic Sequence: The Formula

 

If you wish to find any term (also known as the nth term) in the arithmetic sequence, the arithmetic sequence formula should help you to do so. The key step is to be able to identify or extract known values from the problem that will eventually be substituted into the formula itself.

Let's start by examining the basic parts of the formula:

 

Arithmetic Sequence Formula
the formula to find the nth term of an arithmetic sequence

Where:

a sub n = the term that you want to find
a sub 1 = first term in the list of ordered numbers
n = the term position (ex: for 5th term, n = 5 )
d = common difference of any pair of consecutive numbers

 

Let's put this formula in action!


 

Example 1: Find the 35th term in the arithmetic sequence 3, 9, 15, 21, ...

 

There are three things needed in order to find the 35th term using the formula:

  • the first term ( a1 )
  • the common difference between consecutive terms ( d )
  • and the term position ( n )

 

From the given sequence, we can easily read off the first term and common difference. The term position is just the "n" value in the nth term, thus in 35th term, n=35.

 

common difference of the sequence 3,9,15,21,... is 6

 

Therefore, the known values that we will substitute in the arithmetic formula are

first term = 3, common difference=6 and term position = 35

 

 

 

 

So the solution to find the missing term is,

a sub n = a1 + (n-1)*d, a sub 35 = 207

 

 

 

 

 

 


 

 

Example 2: Find the 125th term in the arithmetic sequence 4, −1, −6, −11, ...

 

This arithmetic sequence has the first term a1= 4, and a common difference of −5.

 

first term = 4 and common difference is -5

 

Since we want to find the 125th term, the "n" value would be n = 125. The following are the known values we will plug into the formula:

 

 

 

 

The missing term in the sequence is calculated as,

a sub 125 = -616

 

 

 

 

 

 

 


 

 

Example 3: If one term in the arithmetic sequence is a sub 21 = -17 and the common difference is d=-3. Find the following:

a) Write a rule that can find any term in the sequence.

b) Find the twelfth term (a sub 12) and eighty-second term (a sub 82) term.

 

 

Solution to part a)

Because we know a term in the sequence which is a sub 21 = -17 and the common difference d=-3, the only missing value in the formula which we can easily solve is the first term a sub 1.

an=a1+(n-1)d , -17=a1+(21-1)*(-3), a1=43

 

Since a1=43 and d=-3, the rule to find any term in the sequence is

an=43+(n-1)*(-3)

 

 

 

 

How do we really know if the rule is correct? What I would do is verify it with the given information in the problem that a21=-17.

So we ask ourselves, what is a21 = ?

We already know the answer though but we want to see if the rule would give us −17.

Since a1= 43, n = 21 and d = −3

a21=43+(21-1)*(-3)=43+(-60)=-17

 

 

 

 

 

 

 

Which it does! Great.

 

Solution to part b)

To answer the second part of the problem, use the rule that we found in part a) which is

close form of arithmetic sequence formula

Here are the calculations side-by-side.

Finding the twelfth term (a12) Finding the eighty-second term (a82)
a12=10 a82=-200

 

 


 

Example 4: Given two terms in the arithmetic sequence, a5=-8 and a25=72,

a) Write a rule that can find any term in the sequence.

b) Find the 100th term (a100).

 

 

Solution to part a)

The problem tells us that there is an arithmetic sequence with two known terms which are a5=-8 and a25=72. The first step is to use the information of each term and substitute its value in the arithmetic formula. We have two terms so we will do it twice.

 

For term a sub 5 equals -8 , For term a sub 25 = 72,
equation 1: -8=a1+4d equation 2: 72=a1+24d

 

This is wonderful because we have two equations and two unknown variables. We can solve this system of linear equations either by Substitution Method or Elimination Method. You should agree that the Elimination Method is the better choice for this.

Place the two equations on top of each other while aligning the similar terms.

solve two equations simultaneously

 

We can eliminate the term by multiplying Equation # 1 by the number −1 and adding them together.

 

 

Since we already know the value of one of the two missing unknowns which is , it is now easy to find the other value. We can find the value of by substituting the value of on any of the two equations. For this, let's use Equation #1.

a sub 1 = -24

 

After knowing the values of both the first term () and the common difference (), we can finally write the general formula of the sequence.

close form formula: a sub n = -24+(n-1)*4

 

 

Solution to part b)

To find the 100th term (a sub 100) of the sequence, use the formula found in part a)

100th term is equal to 372

 

 

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