How to write a recursive rul for sequences. ex.) 1,3,4,7,11
n= (n-1)+(n-2)
looks like a variation of the Fibonacci numbers.
in this one
Term(n+2) = term(n) + term(n+1),
where term(1) = 1
term(2) = 2
n > 2
To write a recursive rule for sequences, we need to identify the relationship between successive terms in the sequence. In the case of the sequence 1, 3, 4, 7, 11, we notice that each term is obtained by performing a specific operation on the previous term. Let's break it down:
- The first term is 1, so we designate it as the initial condition.
- To find the second term (3), we add 2 to the previous term (1 + 2 = 3).
- The third term (4) is obtained by adding 1 to the previous term (3 + 1 = 4).
- The fourth term (7) is obtained by adding 3 to the previous term (4 + 3 = 7).
- Finally, the fifth term (11) is obtained by adding 4 to the previous term (7 + 4 = 11).
Based on this pattern, we can write a recursive rule for the sequence as follows:
a_n = a_(n-1) + n-1
where a_n represents the nth term in the sequence, and a_(n-1) represents the previous term.
Using this recursive rule, we can calculate any term in the sequence by substituting the appropriate values. For example, to find the 6th term, we would calculate:
a_6 = a_(6-1) + (6-1)
a_6 = a_5 + 5
Now, we can use the formula a_5 = a_4 + 4 to continue substituting the values until we reach the desired term.