Question

patient 4
$n=1$
$n=2$
$n=3$
$f(n) =$ number of smiley faces

Explanation:

Step1: Identify pattern for n=1

Count smileys: $f(1)=3$

Step2: Identify pattern for n=2

Count smileys: $f(2)=6$

Step3: Identify pattern for n=3

Count smileys: $f(3)=10$

Step4: Derive general formula

Recognize triangular number sequence: $f(n)=\frac{(n+1)(n+2)}{2}$

Answer:

The function for the number of smiley faces is $\boldsymbol{f(n)=\frac{(n+1)(n+2)}{2}}$, where $n$ is the given index. For the given values:

• $f(1)=3$
• $f(2)=6$
• $f(3)=10$