Find the 65th term of the arithmetic sequence minus, 10, comma, 9, comma, 28, comma, point, point, .−10,9,28

To find the 65th term of the arithmetic sequence, we need to follow these steps:

1. **Identify the first term (\(a\))**: The first term is given as \(a = -10\).
2. **Find the common difference (\(d\))**: To find the common difference, subtract the first term from the second term:
   \[
   d = 9 - (-10) = 9 + 10 = 19.
   \]

   Verify the difference between the second and third terms:
   \[
   d = 28 - 9 = 19.
   \]
   So, the common difference \(d = 19\).

3. **Use the formula for the \(n\)-th term of an arithmetic sequence**:
   \[
   a_n = a + (n - 1) \cdot d
   \]
   where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

4. **Substitute the known values**:
   \[
   a_{65} = -10 + (65 - 1) \cdot 19
   \]
   \[
   a_{65} = -10 + 64 \cdot 19
   \]
   \[
   a_{65} = -10 + 1216
   \]
   \[
   a_{65} = 1206.
   \]

So, the 65th term of the sequence is \(1206\).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the formula for the sum of the first \(n\) terms in an arithmetic sequence?
2. How do you find the common difference in an arithmetic sequence when terms are not consecutive?
3. What is the 100th term of this arithmetic sequence?
4. Can an arithmetic sequence have a decreasing pattern? If yes, what changes in the formula?
5. How can we check if a given sequence is arithmetic?

**Tip**: Always verify the common difference by checking between multiple consecutive terms to ensure accuracy!

Find the 65th term of the arithmetic sequence: -10, 9, 28, ...

Learn how to find the 65th term of the arithmetic sequence -10, 9, 28... This problem involves the use of the nth term formula for arithmetic sequences and linear patterns. Suitable for students in Grades 9-10.

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