Count the number of dots in each arrangement.
How many dots will be in the sixth triangular
number? . ... ......
Here is why they are called "triangular numbers".
They are the numbers of dots it takes to make
successively bigger triangular arrays of dots.
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. . .
. . . . . .
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. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
We can cheat and count the number of dots in the
sixth array of dots. However that's not what your
teacher had in mind. You are to take the first
three of 1,3,6 dots and figure out a formula
from just those:
The 1st triangle above has
1 dot in the top row and
that's all there is.
So the first triangular number is 1.
The 2nd triangle above has
1 dot in the 1st row and
2 dots in the 2nd row.
That's 1+2 or 3
The 3rd triangle above has
1 dot in the 1st row,
2 dots in the 2nd row, and
3 dots in the 3rd row.
That's 1+2+3 or 6 dots.
So we can see that the nth term
is the sum of the arithmetic
series 1+2+3+ ··· +n
where the difference d is 1.
The formula for the sum of an arithmmtic
series is:
Sn = (a1 + an)
where n=6, a1 = 1, an = n
so the formula for the nth triangular number is
Sn = (1 + n)
So when n = 6:
S6 = (1 + 6) = 3(7) = 21
Now count the dots in the last one and we see that
it does contain 21 dots.
Edwin
