2025 AMC 8 Problems/Problem 4: Difference between revisions

Revision as of 22:17, 24 February 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

~ 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

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

@@ Line 23: / Line 23: @@
 Note that this solution is not practical and very time-consuming.
-~codegirl2013
+~codegirl2013, athreyay
-~athreyay
+== Solution 4 ==
-== 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>.
-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>. To find the answer, we then multiply <math>10</math> by <math>7</math> and subtract that from our 0th term to get to our answer, which is <math>\boxed{\text{(B)\ 37}}</math>.
+~Kapurnicus, NYCnerd
-~Kapurnicus
-~ Minor edit by NYCnerd
 ==  Video Solution 1 ==
-https://youtu.be/rf5c9ulMA2I
+[//youtu.be/rf5c9ulMA2I ~ ChillGuyDoesMath]
-~ ChillGuyDoesMath
+== Video Solution 2 ==
-== Video Solution by SpreadTheMathLove ==
+[//www.youtube.com/jTTcscvcQmI SpreadTheMathLove]
-https://www.youtube.com/watch?v=jTTcscvcQmI
+== Video Solution 3 by Daily Dose of Math ==
-== Video Solution 2 by Daily Dose of Math ==
+[//youtu.be/rjd0gigUsd0 ~Thesmartgreekmathdude]
-https://youtu.be/rjd0gigUsd0
+== Video Solution 4 ==
-~Thesmartgreekmathdude
+[//youtu.be/PKMpTS6b988 Thinking Feet]
-== Video Solution 3 by Thinking Feet ==
+== Video Solution 5 ==
-https://youtu.be/PKMpTS6b988
-== Video Solution 4 ==
-https://youtu.be/VP7g-s8akMY?si=K8Pxs_TQhlR2ntt9&t=211
+[//youtu.be/VP7g-s8akMY?si=K8Pxs_TQhlR2ntt9&t=211 ~hsnacademy]
-~hsnacademy
-== Video Solution 5 by CoolMathProblems ==
+== Video Solution 6 ==
-https://youtu.be/nwUanrEZpcQ
+[//youtu.be/nwUanrEZpcQ CoolMathProblems]
-== Video Solution 6 by Pi Academy ==
+== Video Solution 7 ==
-https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK
+[//youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK Pi Academy]
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