What is an arithmetic sequence, and how do you find its sum 80,19,39,72,60,31,5?

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

### Example of an Arithmetic Sequence:
For example, if you have a sequence like \(2, 5, 8, 11, 14\), the common difference \(d = 3\).

### Sum of an Arithmetic Sequence:
The sum of the first \(n\) terms of an arithmetic sequence can be calculated using the formula:

\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]

Where:
- \(S_n\) is the sum of the first \(n\) terms.
- \(a_1\) is the first term of the sequence.
- \(a_n\) is the \(n\)th (or last) term.
- \(n\) is the number of terms.

### Applying to Your Sequence: 80, 19, 39, 72, 60, 31, 5
First, we check whether the sequence is arithmetic by finding the difference between consecutive terms:

\[
\begin{aligned}
19 - 80 &= -61, \\
39 - 19 &= 20, \\
72 - 39 &= 33, \\
60 - 72 &= -12, \\
31 - 60 &= -29, \\
5 - 31 &= -26.
\end{aligned}
\]

Since the differences are not constant, this sequence is **not** arithmetic.

### Conclusion:
Since the sequence \(80, 19, 39, 72, 60, 31, 5\) is not arithmetic, you cannot use the arithmetic sequence sum formula. Instead, you can find the sum by directly adding the terms:

\[
\text{Sum} = 80 + 19 + 39 + 72 + 60 + 31 + 5 = 306
\]

Would you like more details, or do you have any questions? Here are some related questions:

1. How can we determine if a sequence is geometric?
2. What is a geometric sequence, and how do you find its sum?
3. How do you find the sum of an arithmetic sequence with missing terms?
4. What is the difference between an arithmetic and geometric sequence?
5. How would you find the sum of an infinite series?

**Tip:** If you identify a sequence as arithmetic, the common difference should be consistent throughout. Always verify this before using the arithmetic sequence sum formula.

Learn how to determine if a sequence is arithmetic and how to find the sum of non-arithmetic sequences like 80, 19, 39, 72, 60, 31, 5. This problem explores the concept of arithmetic sequences and provides step-by-step guidance on calculating the sum. Suitable for high school students.

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