Arithmetic sequence with explanation 86, _, 45, _

To solve the problem of finding the missing terms in the arithmetic sequence \(86, \_, 45, \_\), we first need to determine the common difference \(d\) of the sequence.

### Understanding Arithmetic Sequences

An **arithmetic sequence** is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the **common difference** (\(d\)).

### Given Sequence

In this problem, we have the sequence: 

\[ 
86, \_, 45, \_ 
\]

### Finding the Common Difference

To find the common difference, let's set up the general formula for the terms of an arithmetic sequence:

1. \(a_1 = 86\) (the first term)
2. \(a_3 = 45\) (the third term)

Using the formula for the \(n\)-th term of an arithmetic sequence:

\[
a_n = a_1 + (n-1) \times d
\]

We can write the equation for the third term:

\[
a_3 = a_1 + 2d = 45
\]

Substitute \(a_1 = 86\) into the equation:

\[
86 + 2d = 45
\]

Solving for \(d\):

\[
2d = 45 - 86
\]
\[
2d = -41
\]
\[
d = \frac{-41}{2} = -20.5
\]

### Finding the Missing Terms

1. **Second Term (\(a_2\))**:

   Using the formula for the second term:

   \[
   a_2 = a_1 + d = 86 - 20.5 = 65.5
   \]

2. **Fourth Term (\(a_4\))**:

   Using the formula for the fourth term:

   \[
   a_4 = a_3 + d = 45 - 20.5 = 24.5
   \]

### Complete Sequence

Thus, the complete arithmetic sequence is:

\[
86, 65.5, 45, 24.5
\]

### Conclusion

The missing terms in the arithmetic sequence are \(65.5\) and \(24.5\).

Would you like more details on any step? Feel free to ask questions!

### Further Exploration

1. What is the formula for the \(n\)-th term of an arithmetic sequence?
2. How can we find the sum of the first \(n\) terms in an arithmetic sequence?
3. What distinguishes arithmetic sequences from geometric sequences?
4. How do we determine the common difference if three terms are given?
5. Can an arithmetic sequence have a common difference of zero? What does it look like?
6. How are arithmetic sequences used in real-life applications?
7. What happens to the terms of an arithmetic sequence if the common difference is negative?
8. Can the terms of an arithmetic sequence be fractions or decimals?

### Tip

When dealing with sequences, always double-check the calculation of the common difference, as it is key to finding the missing terms correctly.

Learn how to solve the problem of finding missing terms in the arithmetic sequence 86, _, 45, _. This problem involves determining the common difference and using the formula for arithmetic sequences. Suitable for students in Grades 7-9.

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