Answer:
- a[n] = 3n+7 . . . . . explicit
- a[1] = 10; a[n] = a[n-1]+3 . . . . recursive
Explanation:
1. The explicit formula for an arithmetic sequence is ...
a[n] = a[1] + (n-1)d
where a[1] is the first term and d is the common difference.
Your sequence has a[1] = 10 as the first term. The common difference is ...
d = 13-10 = 3
Putting these numbers into the explicit formula gives ...
a[n] = 10 + (n-1)·3
This can be simplified to ...
a[n] = 3n +7
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2. The whole point of an arithmetic sequence is that the term of interest is the previous term plus the common difference. Then the recursive formula for an arithmetic sequence is ...
a[n] = a[n-1] +d
A recursive formula also requires the initial value(s) be defined. a[1] is always the first term of the sequence. Using the numbers from above, we have ...
a[1] = 10; a[n] = a[n-1] +3
_____
Comment on the formulas
The explicit formula shows up in algebra problems often enough that it can be useful to memorize it or have it in a handy reference location.
The recursive formula falls out of the definition of an arithmetic sequence (sequential terms have a common difference), so no particular brain cells need to be utilized to remember it.