We can begin on any number we like, so long as we are sure to subtract by 3 for each new term.
Example Sequence 1: {12,9,6,3} Sequence 1 Equation: a_n=-3n+15 Example Sequence 2: {1,-2,-5,-8} Sequence 2 Equation: a_n=-3n+4
Practice makes perfect
There are infinitely many possibilities for these sequences as we can begin on any number we like. The only restriction is that the common difference d must be -3, so we have to subtract 3 to find each consecutive term. Here are two example sequences.
Example1:& 12-3 →9-3 →6-3 →3
Example2:& 1-3 →-2-3 →-5-3 →-8Let's look at these sequences individually in order to find their equations.
Example 1
The first four terms of the first sequence we will be looking at are 12, 9, 6, and 3. They have a common difference d of -3 and the first term a_1 is 12. Let's recall the general rule for an arithmetic sequence.
a_n=a_1+(n-1)d
Substituting our values into the formula gives us the following equation.
a_n=12+(n-1)(-3)
We can rewrite this equation into something resembling slope-intercept form to make it easier to solve for other terms.
The first four terms of the second sequence we will be looking at are: 1, -2, -5, -8. Here we have a common difference d of -3 and the first term a_1 is 1. Using the general form for an arithmetic sequence, we get:
a_n=1+(n-1)(-3)
We can rewrite this equation into something like slope-intercept form to make it easier to solve for other terms.
