Difference between revisions of "2016 AMC 10B Problems/Problem 18"
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<math>(-344,690)</math>,<math>(-108,210)</math>,<math>(-66,138)</math>,<math>(-15,46)</math>,<math>(-3,30)</math>,<math>(30,10)</math>,<math>(50,6)</math>,<math>(172,2)</math> | <math>(-344,690)</math>,<math>(-108,210)</math>,<math>(-66,138)</math>,<math>(-15,46)</math>,<math>(-3,30)</math>,<math>(30,10)</math>,<math>(50,6)</math>,<math>(172,2)</math> | ||
Taking out our erroneous solutions: | Taking out our erroneous solutions: | ||
| − | <math>(30,10),< | + | <math>(30,10)</math>,<math>(50,6)</math>,<math>(172,2)</math> |
Which also gives us three ways. | Which also gives us three ways. | ||
| − | Counting our cases: < | + | Counting our cases: <math>3</math>+<math>3</math>=<math>6</math>,<math>D</math> |
Revision as of 12:13, 21 February 2016
Problem
In how many ways can 
be written as the sum of an increasing sequence of two or more consecutive positive integers?
Solution
Let us have two cases, where 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 
, and the number of terms 
.
We can break down into 
. Thus our possible 
are the following: 
,
,
,
,
(69,5),
,
However, the erroneous solutions which have negatives or have only 1 term are the following:
,,,,
So, we are left with:
(69,5)and
Which gives us three possible ways.
Case 2: Sum of increasing even number of numbers:
If the first term is and there are numbers, this is the following sum:
In this case we have our following (x,n):
,
,
,
,
,
,
,
Taking out our erroneous solutions:
,,
Which also gives us three ways.
Counting our cases: 
+=
,
