The explicit rule of an arithmetic sequence combines the information provided by the two equations of the recursive rule into a single equation.
Recursive:& a_n=a_(n-1)+ d;
& a_1= a_1 [0.8em]
Explicit:& a_n= a_1+(n-1) d
In both of these rules, d is the common difference and a_1 is the first term. Let's start by substituting 1 for n in the given rule in order to find the value of a_1.
We know that a_1= 7, so now we need to identify the value of d. Let's take a look at the given rule.
a_n=3+4n
Notice that the coefficient of n is 4, so we know that the terms are increasing by 4 each time. For the n^(th) term, the number of times it has increased in relation to a_1 is given by the value of n. Therefore, the common difference d is 4. Now we have enough information to form a recursive rule for our sequence.
a_n= & a_(n-1)+ 4;
a_1= & 7 [0.8em]
Therefore, the recursive rule for our sequence is a_1=7, a_n=a_(n-1)+4.
