Does the difference between any two consecutive terms always have the same value?
Recursive Rule: f(1)=2, f(n)=f(n-1)+2, for n≥ 2 Explicit Rule: f(n)=2+2(n-1), for n≥ 1
Practice makes perfect
We want write the recursive rule and the explicit rule for the given arithmetic sequence. Let's pay close attention to the difference between two consecutive terms.
2 +2 → 4 +2 → 6 +2 → 8 +2 → 10...
We can see that the difference between any two consecutive terms is 2.
Recursive Rule
Let's now consider the general formula for a recursive rule.
&f(1)=a
&f(n)=f(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 2 and the common difference is 2.
&f(1)= 2
&f(n)=f(n-1)+ 2,forn≥ 2
Explicit Rule
Finally, let's recall the general formula for an explicit rule.
f(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= 2 and d= 2.
f(n)= 2+ 2(n-1),forn≥ 1
