We want $25$ to be the sixth term of our arithmetic sequence. Recall that the difference between consecutive terms is the common difference $d .$

Since we do not have any restrictions for the common difference, we can arbitrarily choose any value. Let the common difference be

d = 4 .

Knowing the sixth term and the common difference, we can find the fifth term. To do so, we subtract the common difference from the value of the sixth term.

A (5) = 25 - 4 \Leftrightarrow A (5) = 21

In a similar way, we can find

A (4),

A (3),

A (2),

and

A (1) .

A (4) = 21 - 4 A (3) = 17 - 4 A (2) = 13 - 4 A (1) = 9 - 4 \Leftrightarrow A (4) = 17 \Leftrightarrow A (3) = 13 \Leftrightarrow A (2) = 9 \Leftrightarrow A (1) = 5

With this information, we can complete our table.

Now we can use the explicit formula for an arithmetic sequence to write a function rule for our sequence.

A (n) = A (1) + (n - 1) d

In this formula,

A (1)

is the first term and

d

is the common difference. The first term of our sequence is

5,

and the common difference is

4 .

We can substitute these values into the above formula.

A (n) = 5 + (n - 1) 4

This is the function rule we needed. Note that there are infinitely many solutions to this exercise, we illustrated only one example.

Checking Our Answer

Substituting

6

Into the Function Rule

Let's check if the function rule describes a sequence that has

25

as the sixth term. To do so, we will substitute

6

for

n

into the function rule.

A (n) = 5 + (n - 1) 4

Substitute

$n = 6$

A (6) = 5 + (6 - 1) 4

▼

Evaluate right-hand side

SubTerm

Subtract term

A (6) = 5 + (5) 4

Multiply

A (6) = 5 + 20

AddTerms

Add terms

A (6) = 25

As we can see, the sixth term of the sequence represented by the function rule is

25 .

Therefore, we are correct.

Exercise

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