From the table shown in the image, we can observe the number of boxes in each stage of the pattern as follows:

• Stage 1: 4 boxes
• Stage 2: 7 boxes
• Stage 3: 10 boxes
• Stage 4: 13 boxes

The first question asks us to determine the number of boxes in stage 10, and the second question asks us to write an expression for the number of boxes in stage $n$.

Part a) Finding the number of boxes in stage 10:

Looking at the difference between the number of boxes in each stage, we notice that the pattern increases by 3 boxes in each subsequent stage:

• From Stage 1 to Stage 2: $7 - 4 = 3$
• From Stage 2 to Stage 3: $10 - 7 = 3$
• From Stage 3 to Stage 4: $13 - 10 = 3$

Thus, the difference between the number of boxes is constant, indicating this is an arithmetic sequence where the common difference $d = 3$.

The number of boxes in an arithmetic sequence can be given by the formula: $a_n = a_1 + (n-1) \cdot d$ where:

• $a_n$ is the number of boxes at stage $n$,
• $a_1$ is the number of boxes in stage 1 (which is 4),
• $d$ is the common difference (which is 3),
• $n$ is the stage number.

For stage 10, we substitute $n = 10$, $a_1 = 4$, and $d = 3$ into the formula: $a_{10} = 4 + (10 - 1) \cdot 3 = 4 + 9 \cdot 3 = 4 + 27 = 31$

So, the number of boxes in stage 10 is 31.

Part b) Expression for the number of boxes in stage $n$:

Using the same arithmetic sequence formula: $a_n = a_1 + (n - 1) \cdot d$ Substituting $a_1 = 4$ and $d = 3$, we get: $a_n = 4 + (n - 1) \cdot 3 = 4 + 3n - 3 = 3n + 1$

Thus, the expression for the number of boxes in stage $n$ is: $a_n = 3n + 1$

Summary:

• The number of boxes in stage 10 is 31.
• The expression for the number of boxes in stage $n$ is $3n + 1$.

Do you have any questions or want further details?

Here are 5 related questions:

1. What is the number of boxes in stage 20 using the expression $3n + 1$?
2. How can we generalize the formula if the common difference changes?
3. Can you calculate the number of boxes in stage 0 using the same formula?
4. How would the pattern change if the common difference was 5 instead of 3?
5. What type of sequence is this, and how does it differ from geometric sequences?

Tip: When working with arithmetic sequences, always check if the common difference is constant between terms to confirm the sequence type.