Definition
A recursive formula for a sequence \(a_1, a_2, a_3, a_4, a_5,\ldots\) is a formula for computing \(a_n\) using previous terms of the sequence as well as the variable $n$. Examples of recursive formulas are \(a_n = 2a_{n-1} + 5a_{n-2}\) or \(a_n = 3a_{n-1} + n\).
An explicit formula for \(a_n\) only uses the variable \(n\). An example of an explicit formula is \(a_n = n+ 2^n\).
Exercise
Each sequence below is described with a recursive formula. Write out the first six terms of each sequence.
-
\(a_n = 2a_{n-1} + 1\) with \(a_1 = 3\)
-
\(a_n = a_{n-1} + n\) with \(a_1 = 7\)
-
\(a_n = a_{n-1} + a_{n-2}\) with \(a_1 = 4\) and \(a_2 = 7\)
-
\(a_n = a_{n-1} \cdot a_{n-2}\) with \(a_1 = 2\) and \(a_2 = 3\)
Exercise
Suppose a sequence has a recursive formula \(a_n = a_{n-1} + 7\) with \(a_1 = 4\). Write out the first 5 terms of the sequence. Do you recognize it as being arithmetic or geometric? Write an explicit formula for the sequence.
Exercise
Suppose a sequence has a recursive formula \(a_n = 5a_{n-1} - 6a_{n-2}\) with \(a_1 = 2\) and \(a_2 = 5\). Write out the first 5 terms of the sequence. Confirm that an explicit formula for the sequence is given by \(a_n = 2^{n-1} + 3^{n-1}\).
Exercise
Consider the sequence \(2,6,12,20,30,\ldots\).
-
What is the pattern? Use it to fill in the table and find a recursive formula for the sequence.
Term Work to compute the term from previous terms Value of the term \(a_1\) N/A 2 \(a_2\) \(a_2 = a_1 + 4\quad\quad\) 6 \(a_3\) \(a_3 = \phantom{a_1 + 4}\quad\quad\) 12 \(a_4\) \(a_4 = \phantom{a_1 + 4}\quad\quad\) 20 \(a_5\) \(a_5 = \phantom{a_1 + 4}\quad\quad\) 30 \(\vdots\) \(\vdots\) \(\vdots\) \(a_n\) \(a_n = \phantom{a_1 + 4}\quad\quad\) N/A -
Now look at the number of blocks in the sequence of figures below. What do you notice? Use this to find an explicit formula for the sequence above.