Does the difference between any two consecutive terms always have the same value?
Recursive Rule: A(1)=18, A(n)=A(n-1)-7, for n≥ 2 Explicit Rule: A(n)=18+(-7)(n-1), for n≥ 1
Practice makes perfect
We want to determine whether the sequence is an arithmetic sequence and, if so, write its recursive rule and explicit rule. Let's pay close attention to the difference between two consecutive terms.
18 -7 → 11 -7 → 4 -7 → -3 ...
We can see that the difference between any two consecutive terms is -7. Therefore, the given sequence is an arithmetic sequence.
Recursive Rule
Let's now consider the general formula for a recursive rule.
&A(1)=a
&A(n)=A(n-1)+d,forn≥ 2
In the above formula, a is the first term of the sequence and d is the common difference.
For our sequence, the first term is 18 and the common difference is -7.
&A(1)= 18
&A(n)=A(n-1)+( -7),forn≥ 2
Remember that, when adding a negative number, we can rewrite the sum as a difference.
&A(n)=A(n-1) - 7,forn≥ 2
Explicit Rule
Finally, let's recall the general formula for an explicit rule.
A(n)=a+d(n-1),forn≥ 1
Again, a represents the first term of the sequence and d is the common difference. As we have already stated, for our sequence, we have a= 18 and d= -7.
A(n)= 18+( -7)(n-1),forn≥ 1
