Difference between revisions of "2005 PMWC Problems/Problem T6"
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1+2 &=& 3 \\ | 1+2 &=& 3 \\ | ||
4+5+6 &=& 7+8 \\ | 4+5+6 &=& 7+8 \\ | ||
| − | 9+10+11+12 &=& 13+14+15</cmath> | + | 9+10+11+12 &=& 13+14+15\end{eqnarray*}</cmath> |
<cmath>\vdots</cmath> | <cmath>\vdots</cmath> | ||
| − | If this pattern is continued, find the last number in the <math>80</math>th row (e.g. the last number of the third row is <math>15</math>) | + | If this pattern is continued, find the last number in the <math>80</math>th row (e.g. the last number of the third row is <math>15</math>). |
== Solution == | == Solution == | ||
| − | There are 3 numbers in the first row, 5 numbers in the second row, and <math>2n+1</math> numbers in the <math>n</math>th row. Thus for the <math>n</math>th row, the last number is | + | There are 3 numbers in the first row, 5 numbers in the second row, and <math>2n+1</math> numbers in the <math>n</math>th row. Thus for the <math>n</math>th row, the last number is (We use the fact that the sum of the first <math>n</math> odd numbers is <math>n^2</math>): |
<cmath>\left(\sum_{i=1}^{n} 2n+1\right)</cmath> | <cmath>\left(\sum_{i=1}^{n} 2n+1\right)</cmath> | ||
| Line 15: | Line 15: | ||
<cmath>= n^2 + 2n</cmath> | <cmath>= n^2 + 2n</cmath> | ||
| − | + | The last number on the <math>80</math>th row is <math>80^2 + 2\cdot 80 = 6560</math>. | |
== See also == | == See also == | ||
| − | {{PMWC box|year=2005|num-b=T5|num-a= | + | {{PMWC box|year=2005|num-b=T5|num-a=T7}} |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
Latest revision as of 18:07, 10 March 2015
Problem
![\[\vdots\]](http://latex.artofproblemsolving.com/7/6/4/76403adfdd35fb846675aaeaac92daec680a4fba.png)
If this pattern is continued, find the last number in the 
th row (e.g. the last number of the third row is 
).
Solution
There are 3 numbers in the first row, 5 numbers in the second row, and 
numbers in the 
th row. Thus for the th row, the last number is (We use the fact that the sum of the first odd numbers is 
):
![\[\left(\sum_{i=1}^{n} 2n+1\right)\]](http://latex.artofproblemsolving.com/e/a/2/ea2c81d0fb9c62e58c1452689ddbd4d7ca4ec947.png)
![\[= \left(\sum_{i=1}^n 2n-1\right) + 2n\]](http://latex.artofproblemsolving.com/2/1/d/21d98a880ca425750cf45db8410834cb484b3c74.png)
![\[= n^2 + 2n\]](http://latex.artofproblemsolving.com/3/7/b/37b33d3b47c82767c962e35417719fc60d1e3020.png)
The last number on the th row is 
.
See also
| 2005 PMWC (Problems) | ||
| Preceded by Problem T5 |
Followed by Problem T7 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
