Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2025 AMC 8 Problems/Problem 4"
(→‎Video Solution 1)
 
(25 intermediate revisions by 14 users not shown)
Line 1: Line 1:
 +
== Problem ==
 +
 
Lucius is counting backward by <math>7</math>s. His first three numbers are <math>100</math>, <math>93</math>, and <math>86</math>. What is his <math>10</math>th number?
 
Lucius is counting backward by <math>7</math>s. His first three numbers are <math>100</math>, <math>93</math>, and <math>86</math>. What is his <math>10</math>th number?
  
 
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 47</math>
 
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 47</math>
−
==Solution==
 
  
−
By the formula for the <math>n</math>th term of an arithmetic sequence, we get that the answer is <math>a+d(n-1)</math> where <math>a=100, d=-7</math> and <math>n=10</math> which is <math>100 - 7(10 - 1) = \boxed{\text{(B)\ 37}}</math>.
+
== Solution 1 ==
 +
 
 +
We plug <math>a=100, d=-7</math> and <math>n=10</math> into the formula <math>a+d(n-1)</math> for the <math>n</math>th term of an arithmetic sequence whose first term is <math>a</math> and common difference is <math>d</math> to get <math>100-7(10-1) = \boxed{\text{(B)\ 37}}</math>.
  
 
~Soupboy0
 
~Soupboy0
  
−
==Solution 2==
+
== Solution 2 ==
 +
 
 +
Since we want to find the <math>9</math>th number Lucius says after he says <math>100</math>, <math>7</math> is subtracted from his number <math>9</math> times, so our answer is <math>100-(9 \cdot 7) = \boxed{\text{(B)\ 37}}</math>
 +
 
 +
~Sigmacuber
 +
 
 +
== Solution 3 ==
 +
 
 +
Using [[brute force]] and counting backward by <math>7</math>s, we have <math>100, 93, 86, 79, 72, 65, 58, 51, 44, \boxed{\text{(B)\ 37}}</math>.
 +
 
 +
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 <math>100</math>, we have to add <math>7</math> to get to our  0th term, <math>107</math>. Our answer is then <math>107 - 10 \cdot 7 = \boxed{\text{(B)\ 37}}</math>.
 +
 
 +
~Kapurnicus, NYCnerd
 +
 
 +
== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
 +
 
 +
https://www.youtube.com/watch?v=rf5c9ulMA2I
 +
 
 +
~ ChillThingz :)
 +
 
 +
== Video Solution 2 ==
 +
 
 +
[//www.youtube.com/jTTcscvcQmI SpreadTheMathLove]
 +
 
 +
== Video Solution 3 by Daily Dose of Math ==
 +
 
 +
[//youtu.be/rjd0gigUsd0 ~Thesmartgreekmathdude]
 +
 
 +
== Video Solution 4 ==
 +
 
 +
[//youtu.be/PKMpTS6b988 Thinking Feet]
 +
 
 +
== Video Solution 5 ==
 +
 
 +
[//youtu.be/VP7g-s8akMY?si=K8Pxs_TQhlR2ntt9&t=211 ~hsnacademy]
 +
 
 +
== Video Solution 6 ==
 +
 
 +
[//youtu.be/nwUanrEZpcQ CoolMathProblems]
 +
 
 +
== Video Solution 7 ==
 +
 
 +
[//youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK Pi Academy]
 +
==Video Solution(Quick, fast, easy!)==
 +
https://youtu.be/fdG7EDW_7xk
  
−
You could brute force(not really recommended) the sequence, and we get: <cmath>100, 93, 86, 79, 72, 65, 58, 51, 44, 37</cmath> Therefore, our answer is <math>\boxed{\text{(B)\ 37}}</math>
+
~MC
  
−
~athreyay
+
== See Also ==
  
−
==Vide Solution 1 by SpreadTheMathLove==
 
−
https://www.youtube.com/watch?v=jTTcscvcQmI
 
−
==See Also==
 
 
{{AMC8 box|year=2025|num-b=3|num-a=5}}
 
{{AMC8 box|year=2025|num-b=3|num-a=5}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 +
[[Category:Introductory Algebra Problems]]

Latest revision as of 11:45, 7 September 2025

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 (Detailed Explanation) 🚀⚡📊

https://www.youtube.com/watch?v=rf5c9ulMA2I

~ ChillThingz :)

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

Video Solution(Quick, fast, easy!)

https://youtu.be/fdG7EDW_7xk

~MC

See Also

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. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_AMC_8_Problems/Problem_4&oldid=259885"