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How many dots in the nth figure?
Figure 1
One dot
Figure 2
Two dots
Figure 3
Six dots
Figure 4
Ten dots
Example Solution 1:
Example Solution 1:
One of the most common ways to find the nth triangular number is to double the triangle and rearrange it into a rectangle. The rectangle is n units wide and n + 1 units tall. When we find the area of the rectangle, we can divide that by two to count the total number of circles in the triangular number.
Example Solution 2
Example Solution 2
Another way to find the nth triangular number is to reflect the shape about its base to form a square. I've slightly offset the bottom line of the triangle and colored the reflected side so it is easier to see the overlap along the base. This forms a square of side length n. However, since the bottom row has to overlap to form the square, if we add those n circles back, we have exactly twice as many dots as we started with. We can find the total number of circles in the original triangular number by diving the sum of the square and the bottom row by 2.
Example Solution 3:
Example Solution 3:
You may also have noticed the fact that our triangular numbers are made up of sums of consecutive integers.
Check out our solution to the Sum of Integers problem for a strategy often cited as the clever method utilized by mathematician Carl Frederich Gauss when he was in primary school.
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