The explicit formula of an arithmetic sequence combines the information provided by the two equations of the recursive form into a single equation.
Recursive:& a_1= a;
& a_n=a_(n-1)+ d [0.8em]
Explicit:& a_n= a+(n-1) d
In these formulas, d is the common difference and a is the first term. Looking at the given recursive formula, we can identify the common difference d and the value of the first term a.
a_1= 5; a_(n+1)=a_n- 7
Looking at the expression for a_(n+1), we can see that to find a_(n+1), we need to subtract 7 from the previous term a_n. Therefore, we can rewrite the recursive form in using a_n and a_(n-1).
a_(n+1)= a_n- 7 ⇒ a_n = a_(n-1)- 7
Finally, we can see that - 7 is the common difference and the first term is 5. Now we have enough information to form an explicit formula for this sequence.
