A sequence is defined recursively using the formula f(n + 1) = –0.5 f(n) . If the first term of the sequence is 120, what is f(5)?
Answers
Originally Answered: A sequence is defined recursively using the formula f (n + 1) = –0.5 f(n). If the first term of the sequence is 120, what is f(5)?
Your question can be rewritten as:
A sequence is defined recursively as follows:
If n = 1, then f(n) = 120
Else if n > 1, then f(n) =-.5f(n-1) . What is f(5)?
Here are three ways to get the answer:
(I) using hand calculations:
f(1) = 120
f(2) = -.5f(1) = -.5(120) = -60
f(3) = -.5f(2) = -.5(-60) = 30
f(4) = -.5f(3) = -.5(30) = -15
f(5) = -.5f(4) = -.5(-15) = 7.5
Therefore the 5th term of the sequence is 7.5
A derived formula from method (I) is :
f(n) = (-.5)^(n-1)*120 for n > 1.
(II) using the derived formula:
f(5) = (-.5)^4*120 = 7.5
(III) using a python (2.7) program with a recursive function:
print "Program To Find The 5th Term of a Sequence Defined Recursively:"
print " f(1) = 120, and f(n) = -.5f(n-1) for n > 1."
def f(n):
if n == 1:
return 120
else:
if n > 1:
return -.5*f(n-1) # Recursive call to f().
print""
print "Your answer is: ", f(5)
The program output:
Program To Find The 5th Term of a Sequence Defined Recursively:
f(1) = 120, and f(n) = -.5f(n-1) for n > 1.
Your answer is: 7.5
Of course, you can modify the program to find any term of the sequence.
Good luck
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