Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
5. Using Recursive Rules with Sequences
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Exercise 69 Page 450

Practice makes perfect
a First, let's analyze the sequence

Triangular Numbers

The numbers are represented by the points of following diagrams.
Note that the triangular number, is the sum We will find the explicit rule using the formula for finding the sum of an arithmetic series.
For this series, the first term is and the term is Now, let's substitute the values and find the explicit rule for
Therefore, the explicit rule for the sequence of triangular numbers is

Square Numbers

The square number, represents the number of dots in a square with the side made of dots. Therefore, the total number of dots is This tells us that the explicit rule for the sequence of square numbers is

b Of course, the first triangular number is
Notice that the number of dots in is more than the number of dots in Therefore, we can write a recursive rule for
Now, let's deal with the square numbers. The first square number is
Notice that the number of dots in is more than the number of dots in Therefore, we can write a recursive rule for
c Of course Now, let's analyze the situation when Notice that a square can be divide into two smaller triangles, and as in the following picture.

This tells us that for