This problem falls under the subtopic of arithmetic sequences (or arithmetic progressions). In this scenario, the child's allowance increases by a fixed amount each week, making it a classic example of an arithmetic sequence, where:

• The first term is the initial allowance.
• The common difference is the amount of increase each year.

ⓐ Formula for the Child’s Weekly Allowance in a Given Year

The weekly allowance can be modeled using the formula for an arithmetic sequence:

$A_n = A_1 + (n - 1) \cdot d$

Where:

• $A_n$ is the weekly allowance in the $n$-th year.
• $A_1 = 1$ (the initial allowance when the child is 5 years old).
• $d = 2$ (the increase in allowance each year).
• $n$ is the number of years since the child was 5.

Thus, the formula for the child's weekly allowance after $n$ years is:

$A_n = 1 + (n - 1) \cdot 2$

ⓑ Allowance When the Child is 16 Years Old

To find the allowance when the child is 16 years old, we need to determine $n$, which is the number of years after the child turns 5. If the child is 16, then:

$n = 16 - 5 + 1 = 12$

Now, substitute $n = 12$ into the formula:

$A_{12} = 1 + (12 - 1) \cdot 2 = 1 + 11 \cdot 2 = 1 + 22 = 23$

So, the child's weekly allowance when he is 16 years old will be $23.

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Subtopic:

This question falls under arithmetic sequences or arithmetic progressions, which is a common topic in pre-algebra or algebra.

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Would you like a deeper explanation on how arithmetic sequences work, or do you have any other questions on this topic?

Here are some related questions:

1. How would the formula change if the increase was $3 instead of $2?
2. What is the 5th term of this arithmetic sequence?
3. How do you find the sum of the first 10 terms in an arithmetic sequence?
4. What is the common difference in an arithmetic sequence if the first term is $2 and the 5th term is $10?
5. Can this formula be applied to other real-world situations, like saving money or paying off a loan?

Tip: Remember that in arithmetic sequences, the difference between consecutive terms is always the same. This is key to writing the correct formula!