Difference between revisions of "2016 AMC 10B Problems/Problem 18"

(Solution)
(See Also)
Line 34: Line 34:
  
 
==See Also==
 
==See Also==
{{AMC10 box|year=2016|ab=B|num-b=16|num-a=18}}
+
{{AMC10 box|year=2016|ab=B|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:17, 21 February 2016

Problem

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Solution

Let us have two cases, where $345$ is the sum of increasing odd number of numbers and even number of numbers.

Case 1: Sum of increasing odd number of numbers: The mean of an arithmetic sequences with an odd number of numbers is the middle term. Let us call the middle term $x$, and the number of terms $n$.

 $x*n=345$

We can break down $345$ into $3*5*23$. Thus our possible $(x,n)$ are the following: $(1,345)$,$(3,105)$,$(5,69)$,$(15,23)$,$(23,15)$,$(69,5)$,$(105,3)$,$(345,1)$ However, the erroneous solutions which have negatives or have only 1 term are the following: $(1,345)$,$(3,105)$,$(5,69)$,$(15,23)$,$(345,1)$ So, we are left with: $(23,15)$,$(69,5)$ and $(105,3)$ Which gives us three possible ways.

Case 2: Sum of increasing even number of numbers: If the first term is $x$ and there are $n$ numbers, this is the following sum:

$x+x+1+x+2+...+x+(n-1)=$
$xn+1+2+3+...+n-1=$
$xn+n(n-1)/2=$
$n/2(2x+n-1)=$

In this case we have our following $(x,n)$: $(-344,690)$,$(-108,210)$,$(-66,138)$,$(-15,46)$,$(-3,30)$,$(30,10)$,$(50,6)$,$(172,2)$ Taking out our erroneous solutions: $(30,10)$,$(50,6)$,$(172,2)$ Which also gives us three ways. Counting our cases: $3$+$3$=$6$,$D$


See Also

2016 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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. AMC logo.png