Difference between revisions of "2022 AMC 10B Problems/Problem 15"

Revision as of 15:52, 6 June 2024

Problem

Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$ . The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$ . What is $S_{20}$ ?

$\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$

Solution 1

Suppose that the first number of the arithmetic sequence is $a$ . We will try to compute the value of $S_{n}$ . First, note that the sum of an arithmetic sequence is equal to the number of terms multiplied by the median of the sequence. The median of this sequence is equal to $a + n - 1$ . Thus, the value of $S_{n}$ is $n(a + n - 1) = n^2 + n(a - 1)$ . Then, $\frac{S_{3n}}{S_{n}} = \frac{9n^2 + 3n(a - 1)}{n^2 + n(a - 1)} = 9 - \frac{6n(a-1)}{n^2 + n(a-1)}.$ Of course, for this value to be constant, $6n(a-1)$ must be $0$ for all values of $n$ , and thus $a = 1$ . Finally, we have $S_{20} = 20^2 = \boxed{\textbf{(D) } 400}$ .

~mathboy100

Solution 2

Let's say that our sequence is $a, a+2, a+4, a+6, a+8, a+10, \ldots.$ Then, since the value of n doesn't matter in the quotient $\frac{S_{3n}}{S_n}$ , we can say that $\frac{S_{3}}{S_1} = \frac{S_{6}}{S_2}.$ Simplifying, we get $\frac{3a+6}{a}=\frac{6a+30}{2a+2}$ , from which $\frac{3a+6}{a}=\frac{3a+15}{a+1}.$ $3a^2+9a+6=3a^2+15a$ $6a=6$ Solving for $a$ , we get that $a=1$ .

Since the sum of the first $n$ odd numbers is $n^2$ , $S_{20} = 20^2 = \boxed{\textbf{(D) } 400}$ .

Solution 3 (Quick Insight)

Recall that the sum of the first $n$ odd numbers is $n^2$ .

Since $\frac{S_{3n}}{S_{n}} = \frac{9n^2}{n^2} = 9$ , we have $S_n = 20^2 = \boxed{\textbf{(D) } 400}$ .

~numerophile

Video Solution (🚀 Solved in 4 min 🚀)

https://youtu.be/7ztNpblm2TY

~Education, the Study of Everything

Video Solution by Interstigation

https://youtu.be/qkyRBpQHbOA

Video Solution by paixiao

https://www.youtube.com/watch?v=4bzuoKi2Tes

@@ Line 12: / Line 12: @@
 ==Solution 2==
-We'll start with the ratio that term 3 ÷ term 1 = term 6 ÷ term 2
+Let's say that our sequence is <cmath>a, a+2, a+4, a+6, a+8, a+10, \ldots.</cmath>
+Then, since the value of n doesn't matter in the quotient <math>\frac{S_{3n}}{S_n}</math>, we can say that
+<cmath>\frac{S_{3}}{S_1} = \frac{S_{6}}{S_2}.</cmath>
+Simplifying, we get <math>\frac{3a+6}{a}=\frac{6a+30}{2a+2}</math>, from which <cmath>\frac{3a+6}{a}=\frac{3a+15}{a+1}.</cmath> <cmath>3a^2+9a+6=3a^2+15a</cmath> <cmath>6a=6</cmath>
+Solving for <math>a</math>, we get that <math>a=1</math>.
-the sequence goes like: a, a+2, a+4, a+6, a+8, a+10...
+Since the sum of the first <math>n</math> odd numbers is <math>n^2</math>, <math>S_{20} = 20^2 = \boxed{\textbf{(D) } 400}</math>.
-term 3 ÷ term 1 = a+4 ÷ a
-term 6 ÷ term 2 = a+10 ÷ a+2
-a+4 ÷ a = a+10 ÷ a+2
-(a+4)(a+2) = (a)(a+10)
-a^2+6a+8 = a^2+10a
-a+8=10a
-=4a
-=a
-the sequence is updated to 2,4,6,8,10,12...40
-or 2(1+2+3+4+5+6...+20)
-which is also 2(20 x 21)/2 or 20 x 21 and that is 420.
 ==Solution 3 (Quick Insight)==

2022 AMC 10B (Problems • Answer Key • Resources)
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