a Let's draw the next few figures in the pattern and count the total number of triangles in each Figure.
Counting the triangles, we can get the following results.
Figure
Number of Triangles
1
1
2
4
3
10
4
19
5
31
6
46
b Let a_n be the number of triangles in Figure n. From the figures from Part A we can see that the number a_(n+1) is 3n bigger than a_n.
Figure
Number of Triangles
1
a_1=1
2
a_2=4=1+3* 1=a_()darkorange1+3* 1
3
a_3=10=4+3* 2=a_()darkorange2+3* 2
4
a_4=19=10+3* 3=a_()darkorange3+3* 3
5
a_5=31=19+3* 4=a_()darkorange4+3* 4
6
a_6=46=31+3* 5=a_()darkorange5+3* 5
...
...
n+1
a_(n+1)=a_()darkorangen+3n
Since a_1=1, we can write a recursive rule for a_n.
Recursive Rule:
a_1=1, a_(n+1)=a_n+3nforn≥ 1
c We are asked to find a_(10), the number of triangles in Figure 10. Let's use the recursive formula from Part B to find it.
n
a_(n+1)=a_n+3n
a_n+3n
a_(n+1)
0
a_1=1
-
a_1= 1
1
a_2= a_1+3* 1
1+3* 1
a_2= 4
2
a_3= a_2+3* 2
4+3* 2
a_3= 10
3
a_4= a_3+3* 3
10+3* 3
a_4= 19
4
a_5= a_4+3* 4
19+3* 4
a_5= 31
5
a_6= a_5+3* 5
31+3* 5
a_6= 46
6
a_7= a_6+3* 6
46+3* 6
a_7= 64
7
a_8= a_7+3* 7
64+3* 7
a_8= 85
8
a_9= a_8+3* 8
85+3* 8
a_9= 109
9
a_(10)= a_9+3* 9
109+3* 9
a_(10)=136
Therefore, the number of triangles in Figure 10 is 136.
Extra
Calculations Using a Spreadsheet
Let's enter n in cell A1, a(n+1) in cell B1, 0 in cell A2, and a_1=1 in cell B2. Then, write =A2+1 in cell A3. When we hit enter, we will get the following.
Next, copy cell A3, highlight cells A4 through A11, and paste.
Now, write =B2+3*A3 in cell B3. When we hit enter, we will get the following.
Next, copy cell B3, highlight cells B4 through B11, and paste.
Therefore, from the 11^(th) row in the spreadsheet we can get that a_(10)=136. This tells us that the number of triangles in Figure 10 is 136.
