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

Revision as of 18:54, 10 March 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 $\boxed{\textbf{(B)}\ 21}$ .

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 18: / Line 18: @@
 ==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 <math>1 , 4 , 7,..., 61</math>.
--1/3=20 20+1=21
+<math>61-1/3=20, 20 + 1 = 21</math>
 <math>\boxed{\textbf{(B)}\ 21}</math>.

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