Pearson Algebra 1 Common Core, 2011
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Pearson Algebra 1 Common Core, 2011 View details
7. Arithmetic Sequences
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Exercise 38 Page 279

Can you find the common difference and the first term just by looking at the recursive formula?

Practice makes perfect
The explicit formula of an arithmetic sequence combines the information provided by the two equations of the recursive form into a single equation.
In these formulas, is the common difference and is the first term. Looking at the given recursive formula, we can identify the common difference and the value of the first term
We can see that is the common difference and the first term is also Now we have enough information to form an explicit formula for this sequence.

Extra

The Fibonacci Sequence

One particularly well-known sequence that is defined recursively is the Fibonacci sequence, in which each term is the sum of the two previous terms. It is a well-known sequence whose first few terms are As shown, the first two terms are and and each subsequent term will be the sum of the previous two terms.

Fibonacci numbers in the Fibonacci sequence
For every term after the second term, the sequence can be expressed by the following recursive rule.
This means that, in order to determine the fourth term of the sequence, , one must have the terms and .
The sequence is named after the Italian mathematician Leonardo Fibonacci, who used it to describe how pairs of rabbits increases as they multiply under certain conditions. Sometimes, the first two numbers are defined as and This, however, does not change the sequence in any way other than by increasing the index of every term by one compared to the previously mentioned definition.