1.
Explore the first sequence of dots from the Investigate! activity. We will let \(a_n\) represent the number of dots in figure \(n\text{.}\) The sequence starts \(1, 5, 9,\ldots\)
(a)
How many dots would you expect in the next two figures in the sequence?
Dots in \(n = 3\) figure: \(a_3 =\).
Dots in \(n = 4\) figure: \(a_4 =\).
(b)
How is the sequence growing? To get the next figure, take the current figure and
- add
- multiply by
the constant dots.
(c)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
\(a_n =\) ; with \(a_0 =\) .
(d)
Guess a closed formula for the number of dots in the \(n\)th figure.
\(a_n =\) .
