2025 AMC 8 Problem 4

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To help you master 2025 AMC 8 Problem 4, this page includes:

The original AMC problem to try, plus the correct answer

Clear, step-by-step solutions using different methods

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2025 AMC8 Problem 4

Solution

Solution 1

Jump Calculation Method

Instead of subtracting 7 step by step, we can group all the subtractions together. To reach the 10th number from the 1st number, we need to take exactly 9 jumps.

The number of jumps between the 1st item and the $n$ th item in a sequence is $n - 1$ .

Step 1: Identify the starting number and the rule

We establish the beginning of our sequence and the constant amount by which it changes at each step.

First number = 100

Rule = - 7 per step

Step 2: Determine the number of jumps

Because the 1st number is our starting point (0 jumps), reaching the 2nd number requires 1 jump, the 3rd requires 2 jumps, and so on. We calculate the number of jumps to reach the 10th number.

Number of jumps = 10 - 1 = 9

Step 3: Calculate the total change

We multiply the number of jumps by the size of each jump to find out how much the sequence will decrease in total.

Total change = 9 \times 7 = 63

Step 4: Calculate the 10th number

We subtract the total change from our starting number to find the final result.

100 - 63 = 37

Correct answer: B. $37$
The 10th number is 37, which matches answer choice B. We can quickly verify by estimation: 10 steps of 7 would drop the number by 70 to 30, so 9 steps of 7 should drop it slightly less, landing in the upper 30s. This makes 37 a very reasonable answer.

Solution 2

Direct Listing Method

For a sequence with a small number of terms, we can simply keep subtracting 7 step by step until we reach the target position. This is a very safe way to avoid counting errors.

Counting directly guarantees we do not make the ‘fencepost error’ (being off by one step).

Step 1: Record the initial numbers given in the problem

We write down the first three terms to establish our numbered list.

1st : 100

2nd : 93

3rd : 86

Step 2: Subtract 7 repeatedly until reaching the 10th number

We continue applying the rule one step at a time, keeping careful track of each term’s position.

4th : 86 - 7 = 79

5th : 79 - 7 = 72

6th : 72 - 7 = 65

7th : 65 - 7 = 58

8th : 58 - 7 = 51

9th : 51 - 7 = 44

10th : 44 - 7 = 37

Correct answer: B. $37$
The 10th number is 37 (Choice B). Because we explicitly listed all 10 numbers and verified every subtraction, we can be completely confident in the result.

Solution 3

Zeroth Term (Linear Function) Method

A sequence that changes by a constant amount acts like a line on a graph. If we work backward to find a hypothetical ‘0th’ term, we can find any term simply by multiplying the step number by the jump size.

By finding the 0th term, the $n$ th term becomes directly proportional to $n$ , eliminating the need to subtract 1 from the term number.

Step 1: Find the hypothetical 0th number

The sequence decreases by 7 as we go forward. If we go backward one step from the 1st number, we must add 7. This gives us the starting value at ‘step zero’.

0th number = 100 + 7 = 107

Step 2: Create a direct formula for the nth number

Now we can think of the sequence as starting at the 0th number and subtracting 7 exactly $n$ times to reach the $n$ th number.

nth number = 107 - 7 n

Step 3: Calculate the 10th number

We substitute 10 for $n$ into our rule to find the answer directly.

10th number = 107 - 7 \times 10 = 107 - 70 = 37

Correct answer: B. $37$
The result is 37 (Choice B). We can check our rule on the 1st term:

107 - 7 (1) = 100

. Since it correctly produces the known 1st term, we know the rule is valid.

Adapted Problems to Practice

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What Formulas Do You Need for This Problem?

Arithmetic Sequence $n$ th Term Formula:

a_{n} = a_{1} + (n - 1) d

$a_{n}$ is the $n$ th term, $a_{1}$ is the first term, $d$ is the constant difference, and $n - 1$ is the number of jumps. Because the sequence changes by the same constant amount ( $d = - 7$ ) at each step, we can jump directly to any term without listing all the intermediate numbers.

The Zeroth Term Strategy:

nth term = 0th term + n \times d

The '0th term' is found by working backward one step from the 1st term. $n$ is the term number, and $d$ is the constant change per step. An arithmetic sequence acts just like a linear equation ( $y = m x + b$ ). Finding the 0th term provides the y-intercept ( $b$ ), making the calculation simpler by matching the jump multiplier directly to the term number.

Key Takeaway: Remember This Strategy for Next Time

A sequence changes by the exact same amount at each step.

Try: Count the number of jumps (which is always one less than the number of terms) and multiply by the jump size.

Why: The first term is your starting point, requiring 0 jumps. To reach the $n$ th term, you only make $n - 1$ jumps. Remembering this avoids the very common 'fencepost error'.

You need to find a distant term in a sequence and want to avoid the $n - 1$ calculation.

Try: Find the hypothetical '0th' term by taking one step backward from the 1st term. Then, the $n$ th term is simply the 0th term plus $n$ jumps.

Why: This shifts the starting point so that the term number matches the number of jumps exactly, transforming the sequence into a simple linear function.

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2025 AMC 8 Problem 4

In This Page:

2025 AMC8 Problem 4