https://file.aiquickdraw.com/mathbot/file/172627984404579j0xcgt.jpg

Let's break down and solve each part of the question as shown in the image.

### (a) Write down an expression for the nth term of the sequence:
The sequence given is: \(1, -4, -9, -14, -19\).
This appears to be an arithmetic sequence where each term is decreasing by a constant value. The common difference \(d\) is:
\[
d = -4 - 1 = -5
\]
The first term \(a_1 = 1\). The general formula for the nth term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1) \cdot d
\]
Substitute \(a_1 = 1\) and \(d = -5\):
\[
a_n = 1 + (n-1)(-5)
\]
Simplify:
\[
a_n = 1 - 5n + 5 = -5n + 6
\]
So, the nth term is:
\[
a_n = -5n + 6
\]

### (b) Is \(-110\) a term of this sequence?
We need to determine if \(-110\) can be a value of the nth term, i.e., we solve for \(n\) in:
\[
-5n + 6 = -110
\]
First, subtract 6 from both sides:
\[
-5n = -110 - 6
\]
\[
-5n = -116
\]
Now, divide by \(-5\):
\[
n = \frac{-116}{-5} = 23.2
\]
Since \(n\) is not an integer, \(-110\) is **not** a term of the sequence. Therefore, the answer is **No**.

### (c) Find the value of \(n\) when the sum of the first \(n\) terms is \(-318\):
The formula for the sum of the first \(n\) terms of an arithmetic sequence is:
\[
S_n = \frac{n}{2} [2a_1 + (n-1)d]
\]
Substitute \(S_n = -318\), \(a_1 = 1\), and \(d = -5\):
\[
-318 = \frac{n}{2} [2(1) + (n-1)(-5)]
\]
Simplify:
\[
-318 = \frac{n}{2} [2 - 5n + 5]
\]
\[
-318 = \frac{n}{2} [7 - 5n]
\]
Multiply both sides by 2:
\[
-636 = n(7 - 5n)
\]
Expand the right-hand side:
\[
-636 = 7n - 5n^2
\]
Rearrange to form a quadratic equation:
\[
5n^2 - 7n - 636 = 0
\]
Now, solve this quadratic equation using the quadratic formula:
\[
n = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(5)(-636)}}{2(5)}
\]
\[
n = \frac{7 \pm \sqrt{49 + 12720}}{10}
\]
\[
n = \frac{7 \pm \sqrt{12769}}{10}
\]
\[
n = \frac{7 \pm 113}{10}
\]
So the two possible solutions are:
\[
n = \frac{7 + 113}{10} = 12 \quad \text{or} \quad n = \frac{7 - 113}{10} = -10.6
\]
Since \(n\) must be a positive integer, \(n = 12\).

### Summary of answers:
- (a) The nth term of the sequence is \(a_n = -5n + 6\).
- (b) \(-110\) is not a term of the sequence.
- (c) The value of \(n\) when the sum is \(-318\) is \(n = 12\).

Do you have any questions or want further details on any part?

Here are some related questions for practice:
1. What would the 15th term of this sequence be?
2. How would the sequence change if the first term was 2 instead of 1?
3. How do you derive the formula for the sum of an arithmetic sequence?
4. What if you were asked for the sum of the first 20 terms instead?
5. Can you determine a general rule for checking if a number is part of an arithmetic sequence?

**Tip**: Always check for the common difference first when identifying a sequence type!

The first five terms of a sequence are 1, -4, -9, -14, -19.
(a) Write down an expression for the nth term of the sequence.
(b) Is -110 a term of this sequence?
(c) Find the value of n when the sum of the first n terms is -318.

Learn how to determine the nth term and sum of an arithmetic sequence. This example includes steps for finding the nth term, checking if a number is part of the sequence, and solving a quadratic equation for the sum of the first n terms. Perfect for students in Grades 9-10.

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