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Let's break down the questions and solve them one by one:

### Question 17: 
The sequence given is \( 10, 11, 11, 12, 13, 13, 14, \dots \). The pattern alternates between repeating a number and increasing by 1 after the repetition. Let's observe the pattern:
- 10 appears once, 11 appears twice, 12 appears once, 13 appears twice, 14 appears once, and so on.

So the sequence looks like this: 
\[ 10, 11, 11, 12, 13, 13, 14, \dots \]

To find the nth term of the sequence, notice that for every even position, we have a repeated number, and for every odd position, we have a unique increasing number. Therefore, we can express the sequence based on this alternating pattern.

#### Formula for the nth term:
- If n is odd: The term is \( 5 + \left(\frac{n + 1}{2}\right) \)
- If n is even: The term is \( 5 + \left(\frac{n}{2}\right) \)

### Question 18:
This describes a pile of stones with 10 layers. The bottom layer has 32 stones, and each layer above has fewer stones. Specifically:
- The first layer has 32 stones.
- The second layer has 30 stones.
- The third layer has 28 stones, and so on.

So, the sequence of stones per layer follows the pattern: 32, 30, 28, ...

This is an arithmetic sequence where the first term is \( a_1 = 32 \) and the common difference \( d = -2 \) (since each layer has 2 fewer stones than the one below it).

#### Formula for the nth term of the sequence:
The formula for the nth term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1) \cdot d
\]

In this case:
\[
a_n = 32 + (n-1) \cdot (-2)
\]
\[
a_n = 32 - 2(n-1)
\]
\[
a_n = 32 - 2n + 2
\]
\[
a_n = 34 - 2n
\]

For example:
- The 1st term: \( a_1 = 34 - 2(1) = 32 \)
- The 2nd term: \( a_2 = 34 - 2(2) = 30 \)
- The 3rd term: \( a_3 = 34 - 2(3) = 28 \)
And so on.

Would you like further details on these sequences or additional steps?

Learn how to solve complex sequences, including alternating and arithmetic patterns. The problems focus on finding the nth term of a sequence and applying arithmetic sequence formulas. Suitable for students in Grades 7-9.

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