2025 AMC 8 Problems/Problem 4

Problem

Lucius is counting backward by $7$ s. His first three numbers are $100$ , $93$ , and $86$ . What is his $10$ th number?

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 47$

Solution 1

We plug $a=100, d=-7$ and $n=10$ into the formula $a+d(n-1)$ for the $n$ th term of an arithmetic sequence whose first term is $a$ and common difference is $d$ to get $100-7(10-1) = \boxed{\text{(B)\ 37}}$ .

~Soupboy0

Solution 2

Since we want to find the $9$ th number Lucius says after he says $100$ , $7$ is subtracted from his number $9$ times, so our answer is $100-(9 \cdot 7) = \boxed{\text{(B)\ 37}}$

~Sigmacuber

Solution 3

Using brute force and counting backward by $7$ s, we have $100, 93, 86, 79, 72, 65, 58, 51, 44, \boxed{\text{(B)\ 37}}$ .

Note that this solution is not practical and very time-consuming.

~codegirl2013 ~athreyay

Solution 4

Knowing that our first term is $100$ , we have to add $7$ to get to our "0th term", $107$ . To find the answer, we then multiply $10$ by $7$ and subtract that from our 0th term to get to our answer, which is $\boxed{\text{(B)\ 37}}$ .

~Kapurnicus ~ Minor edit by NYCnerd

Video Solution 1

https://youtu.be/rf5c9ulMA2I

~ ChillGuyDoesMath

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=jTTcscvcQmI

Video Solution 2 by Daily Dose of Math

https://youtu.be/rjd0gigUsd0

~Thesmartgreekmathdude

Video Solution 3 by Thinking Feet

https://youtu.be/PKMpTS6b988

Video Solution 4

https://youtu.be/VP7g-s8akMY?si=K8Pxs_TQhlR2ntt9&t=211 ~hsnacademy

Video Solution 5 by CoolMathProblems

https://youtu.be/nwUanrEZpcQ

Video Solution 6 by Pi Academy

https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK

2025 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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