Express the sequence given below as a recursively-defined function.
3,11,27,59,123
A.
u(0) = 3
u(n + 1) = u(n) + 8
for n = 0, 1, 2, ...
B.
u(0) = 3
u(n + 1) = 2u(n) + 5
for n = 0, 1, 2, ...
C.
u(0) = 3
u(n + 1) = 8u(n) + 1
for n = 0, 1, 2, ...
D.
u(0) = 3
u(n + 1) = 3u(n) + 2
for n = 0, 1, 2, ...
C.
u(0) = 3
u(n + 1) = 8u(n) + 1
for n = 0, 1, 2, ...
The correct answer is B.
The recursively-defined function for the given sequence is:
u(0) = 3
u(n + 1) = 2u(n) + 5
for n = 0, 1, 2, ...
Explanation:
Starting with u(0) = 3, each subsequent term (u(n + 1)) can be obtained by multiplying the previous term (u(n)) by 2 and adding 5. This pattern is followed for all values of n, starting from 0 onwards.
To express the given sequence as a recursively-defined function, we need to find a recursive formula that defines each term in terms of the previous terms.
Let's analyze the given sequence: 3, 11, 27, 59, 123.
Looking at the differences between consecutive terms, we can see that the differences are not constant. Therefore, the recursive formula will not be a simple arithmetic progression.
Now, let's go through the options provided:
A. u(n + 1) = u(n) + 8
This option represents an arithmetic progression with a common difference of 8. However, the given sequence doesn't follow an arithmetic progression.
B. u(n + 1) = 2u(n) + 5
This option represents a sequence where each term is double the previous term plus 5. Let's check if this formula fits the given sequence:
For n = 0, u(0 + 1) = 2u(0) + 5 = 2 * 3 + 5 = 11. It matches the second term.
For n = 1, u(1 + 1) = 2u(1) + 5 = 2 * 11 + 5 = 27. It matches the third term.
For n = 2, u(2 + 1) = 2u(2) + 5 = 2 * 27 + 5 = 59. It matches the fourth term.
For n = 3, u(3 + 1) = 2u(3) + 5 = 2 * 59 + 5 = 123. It matches the fifth term.
B is the correct answer. The recursively-defined function that represents the given sequence is:
u(0) = 3
u(n + 1) = 2u(n) + 5
for n = 0, 1, 2, ...