Find the recursive formula for the following sequence and determine if the sequence is arithmetic or geometric.
A) 3,7,11,15....
B) -5,-15,-75....
A) note that the common difference between terms is 4.
B) The common ratio between terms is 5 if I assume a typo and the first entry is -3.
Otherwise, it is more complicated, and 3 terms don't give much to go on.
To find the recursive formula for a given sequence, we need to identify the pattern or rule governing the sequence. Let's analyze each sequence individually:
A) 3, 7, 11, 15...
Looking at the sequence, we can observe that each number is obtained by adding 4 to the previous number. Therefore, the recursive formula for the sequence is given by:
a(n) = a(n-1) + 4, with a(1) = 3
This recursive formula means that to find each term in the sequence, we add 4 to the previous term starting from the first term of 3.
Now let's determine if the sequence is arithmetic or geometric.
For an arithmetic sequence, the difference between consecutive terms is always constant. In this case, the difference between consecutive terms in the sequence is 4, which is constant. Therefore, the given sequence is arithmetic.
B) -5, -15, -75...
Analyzing this sequence, we can notice that each term is obtained by multiplying the previous term by -5. Therefore, the recursive formula for the sequence is:
a(n) = -5 * a(n-1), with a(1) = -5
This recursive formula means that to find each term in the sequence, we multiply the previous term by -5 starting from the first term of -5.
Now let's determine if the sequence is arithmetic or geometric.
For a geometric sequence, the ratio between consecutive terms is always constant. In this case, the ratio between consecutive terms in the sequence is -5, which is constant. Therefore, the given sequence is geometric.