Answer
$a_1 = 2$
$a_2 = 3$
$a_3 = 5$
$a_4 = 9$
$a_5 = 17$
Work Step by Step
In this exercise, we are given a recursive formula, which relates one term to the term before it.
We are given the initial term $a_1 = 2$, so we need to come up with the next four terms of the sequence, $a_2$ through $a_5$:
For $n = 2$:
$a_2 = 2a_{2 - 1} - 1$
Simplify:
$a_2 = 2a_{1} - 1$
Substitute $a_1 = 1$ into the formula:
$a_2 = 2(2) - 1$
Multiply first:
$a_2 = 4 - 1$
Subtract:
$a_2 = 3$
For $n = 3$:
$a_3 = 2a_{3 - 1} - 1$
Simplify:
$a_3 = 2a_{2} - 1$
Substitute $a_2 = 3$ into the formula:
$a_3 = 2(3) - 1$
Multiply first:
$a_3 = 6 - 1$
Subtract:
$a_3 = 5$
For $n = 4$:
$a_4 = 2a_{4 - 1} - 1$
Simplify:
$a_4 = 2a_{3} - 1$
Substitute $a_3 = 5$ into the formula:
$a_4 = 2(5) - 1$
Multiply first:
$a_4 = 10 - 1$
Subtract:
$a_4 = 9$
For $n = 5$:
$a_5 = 2a_{5 - 1} - 1$
Simplify:
$a_5 = 2a_{4} - 1$
Substitute $a_4 = 9$ into the formula:
$a_5 = 2(9) - 1$
Multiply first:
$a_5 = 18 - 1$
Subtract:
$a_5 = 17$
