Difference between revisions of "2015 AMC 10A Problems/Problem 7"

Latest revision as of 22:05, 26 June 2023

Problem

How many terms are in the arithmetic sequence $13$ , $16$ , $19$ , $\dotsc$ , $70$ , $73$ ?

$\textbf{(A)}\ 20 \qquad\textbf{(B)} \ 21 \qquad\textbf{(C)} \ 24 \qquad\textbf{(D)} \ 60 \qquad\textbf{(E)} \ 61$

Solution 1

$73-13 = 60$ , so the amount of terms in the sequence $13$ , $16$ , $19$ , $\dotsc$ , $70$ , $73$ is the same as in the sequence $0$ , $3$ , $6$ , $\dotsc$ , $57$ , $60$ .

In this sequence, the terms are the multiples of $3$ going up to $60$ , and there are $20$ multiples of $3$ in $60$ .

However, the number 0 must also be included, adding another multiple. So, the answer is $\boxed{\textbf{(B)}\ 21}$ .

Solution 2

Using the formula for arithmetic sequence's nth term, we see that $a + (n-1)d \Longrightarrow13 + (n-1)3 =73, \Longrightarrow n = 21$ $\boxed{\textbf{(B)}\ 21}$ .

Solution 3

Minus each of the terms by $12$ to make the the sequence $1 , 4 , 7,..., 61$ . $\frac{61-1}{3}=20, 20 + 1 = 21$

$\boxed{\textbf{(B)}\ 21}$ .

Solution 4

Subtract each of the terms by $10$ to make the sequence $3 , 6 , 9,..., 60, 63$ . Then divide the each term in the sequence by $3$ to get $1, 2, 3,..., 20, 21$ . Now it is clear to see that there are $21$ terms in the sequence. $\boxed{\textbf{(B)}\ 21}$ .

Video Solution (CREATIVE THINKING)

https://youtu.be/i7ItueJ6K8E

~Education, the Study of Everything

Video Solution

https://youtu.be/fcWPfgKeCmA

~savannahsolver

2015 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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 AMC 10 Problems and Solutions

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

@@ Line 1: / Line 1: @@
 ==Problem==
-How many terms are in the arithmetic sequence 13, 16, 19,..., 70, 73?
+How many terms are in the arithmetic sequence <math>13</math>, <math>16</math>, <math>19</math>, <math>\dotsc</math>, <math>70</math>, <math>73</math>?
-<math> \textbf{(a)}\ 20 \qquad\textbf{(B)} \ 21 \qquad\textbf{(C)} \ 24 \qquad\textbf{(D)} \ 60 \qquad\textbf{(E)} \ 61 </math>
+<math> \textbf{(A)}\ 20 \qquad\textbf{(B)} \ 21 \qquad\textbf{(C)} \ 24 \qquad\textbf{(D)} \ 60 \qquad\textbf{(E)} \ 61 </math>
+==Solution 1==
+<math>73-13 = 60</math>, so the amount of terms in the sequence <math>13</math>, <math>16</math>, <math>19</math>, <math>\dotsc</math>, <math>70</math>, <math>73</math> is the same as in the sequence <math>0</math>, <math>3</math>, <math>6</math>, <math>\dotsc</math>, <math>57</math>, <math>60</math>.
+In this sequence, the terms are the multiples of <math>3</math> going up to <math>60</math>, and there are <math>20</math> multiples of <math>3</math> in <math>60</math>.
+However, the number 0 must also be included, adding another multiple. So, the answer is <math>\boxed{\textbf{(B)}\ 21}</math>.
+==Solution 2==
+Using the formula for arithmetic sequence's nth term, we see that <math>a + (n-1)d \Longrightarrow13 + (n-1)3 =73, \Longrightarrow n = 21</math>
+<math>\boxed{\textbf{(B)}\ 21}</math>.
+==Solution 3==
+Minus each of the terms by <math>12</math> to make the the sequence <math>1 , 4 , 7,..., 61</math>. <math>\frac{61-1}{3}=20, 20 + 1 = 21</math>
+<math>\boxed{\textbf{(B)}\ 21}</math>.
+==Solution 4==
+Subtract each of the terms by <math>10</math> to make the sequence <math>3 , 6 , 9,..., 60, 63</math>. Then divide the each term in the sequence by <math>3</math> to get <math>1, 2, 3,..., 20, 21</math>. Now it is clear to see that there are <math>21</math> terms in the sequence.
+<math>\boxed{\textbf{(B)}\ 21}</math>.
+==Video Solution (CREATIVE THINKING)==
+https://youtu.be/i7ItueJ6K8E
+~Education, the Study of Everything
+==Video Solution==
+https://youtu.be/fcWPfgKeCmA
+~savannahsolver
+==See Also==
+{{AMC10 box|year=2015|ab=A|num-b=6|num-a=8}}
+{{MAA Notice}}

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