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

This can be thought of as an arithmetic sequence. Knowing that our first term is $100$ , we have to add $7$ to get to our 0th term, $107$ . Our answer is then $107 - 10 \cdot 7 = \boxed{\text{(B)\ 37}}$ .

~Kapurnicus, NYCnerd

Video Solution 1

~ ChillGuyDoesMath

Video Solution 2

SpreadTheMathLove

Video Solution 3 by Daily Dose of Math

~Thesmartgreekmathdude

Video Solution 4

Thinking Feet

Video Solution 5

~hsnacademy

Video Solution 6

CoolMathProblems

Video Solution 7

Pi Academy

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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