Question

1. consider this pattern.

which function represents the sequence that represents the pattern?
a. $a_n = a_{n - 1} - 3$
b. $a_n = a_{n - 1} + 3$
c. $a_n = 3a_{n - 1} + 3$
d. $a_n = 3a_{n - 1} - 3$

Explanation:

Step1: Count the number of flowers in each group

First group (left): \(1 + 2 + 3 = 6\) flowers.
Second group (middle): \(2 + 3 + 4 = 9\) flowers.
Third group (right): \(3 + 4 + 5 = 12\) flowers.

Step2: Analyze the sequence

The sequence of the number of flowers is \(6, 9, 12, \dots\). Let's check the recurrence relation.
For \(n = 2\) (second term), \(a_2 = 9\), \(a_1 = 6\). \(9 - 6 = 3\), so \(a_2 = a_1 + 3\).
For \(n = 3\) (third term), \(a_3 = 12\), \(a_2 = 9\). \(12 - 9 = 3\), so \(a_3 = a_2 + 3\).
So the recurrence relation is \(a_n = a_{n - 1} + 3\).

Answer:

b. \(a_{n}=a_{n - 1}+3\)