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

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==Solution 2==
 
==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>
 
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)}}</math>.
+
<math>\boxed{\textbf{(B)}\ 21}</math>.
  
 
==Solution 3==
 
==Solution 3==
 
Minus each of the terms by 12 to make the the sequence 1,4,7,....,61.   
 
Minus each of the terms by 12 to make the the sequence 1,4,7,....,61.   
 
61-1/3=20 20+1=21
 
61-1/3=20 20+1=21
<math>\boxed{\textbf{(B)}}</math>.
+
<math>\boxed{\textbf{(B)}\ 21}</math>.
  
 
==Solution 4==
 
==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
 
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
 +
<math>\boxed{\textbf{(B)}\ 21}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2015|ab=A|num-b=6|num-a=8}}
 
{{AMC10 box|year=2015|ab=A|num-b=6|num-a=8}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:29, 16 January 2020

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

$73-13 = 60$, 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, one more must be added to include the first term. 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. 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 $\boxed{\textbf{(B)}\ 21}$.

See Also

2015 AMC 10A (ProblemsAnswer KeyResources)
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

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