Arithmetic and Geometric Sequences

¶Investigate!18For the patterns of dots below, draw the next pattern in the sequence. Then give a recursive definition and a closed formula for the number of dots in the \(n\)th pattern.We now turn to the question of finding closed formulas for particular types of sequences.Arithmetic SequencesIf the terms of a sequence differ by a constant, we say the sequence is arithmetic. If the initial term (\(a_0\)) of the sequence is \(a\) and the common difference is \(d\text{,}\) then we have,Recursive definition: \(a_n = a_{n-1} + d\) with \(a_0 = a\text{.}\)Closed formula: \(a_n = a + dn\text{.}\)How do we know this? For the recursive definition, we need to specify \(a_0\text{.}\) Then we need to express \(a_n\) in terms of \(a_{n-1}\text{.}\) If we call the first term \(a\text{,}\) then \(a_0 = a\text{.}\) For the recurrence relation, by the definition of an arithmetic sequence, the difference between successive terms is some constant, say \(d\text{.}\) So \(a_n - a_{n-1} = d\text{,}\) or i...