Day3 - Sequences -Recursive Definition
HW#
Name
Date
Per
Find the fourth and fifth terms of each sequence.
1
n
2
1. t1 = 1 ; tn = 3tn−1 − 2
2. t1 = ; tn =
3. t1 = 2 ; t2 = 4 ; tn = (tn−1 − tn−2 )
(tn−1 +1)
2
n +1
4. a. Give the first eight terms of the sequence defined recursively by t1 = 3 , t2 = 5 , and tn = tn−1 − tn−2 .
b. Observing the pattern you get in part (a), tell what the 1000th term of the sequence will be.
1
5. Find an explicit definition for the sequence defined by t1 = 36 ; tn = tn−1
3
Give a recursive definition for each sequence.
6. 5, 7, 11, 19, 35, 67…
7. 2, 6, 24, 120, 720…
8. Suppose a country has a population of 6,000,000, grows 1.5% per year, and that an additional
50,000 people immigrate into the country every year.
a. Give a recursion equation for Pn , the population in n years.
b. If the population is now 8,500,000, what will the population be in 5 years?
9. Let Sn represent the number of dots in an n by n square array. Pretend you have forgotten that
Sn = n 2 . Give a recursion equation that tells how Sn+1 is related to Sn by reasoning how many extra
dots are needed to form the (n + 1)st square array from the previous nth square array. Illustrate your
answer with a diagram of dots. Use a generic picture to show how to get Sn+1 if you have Sn .
10. Visual Thinking Let dn represent the number of diagonals that can be drawn in an n-sided polygon.
A hexagon has 9 diagonals (and six sides, which are not counted), so d6 = 9 .
a. What is the smallest number for n?
b. Find a recursive definition for the number of diagonals in an n-sided polygon.
- To do this, what would George say? First look at the early cases and see if you can generalize it.
To do that, find the first 5 terms. Hint, the first term is not dn , since n = 1 does not make sense.
Look at part a. For each, draw a picture. Now, make a table and write your definition.
c. Here is another way to find the definition. If you look at a suitably general case and find the pattern,
that pattern can sometimes be used to write the definition. Here is an example of it.
i. Imagine starting with a hexagon, which you know has 9 diagonals. If you push out one side of
the hexagon so that a polygon of 7 sides is formed. How many additional diagonals can be drawn?
Draw it. It will be helpful to use different colors to keep clear the sides and new diagonals.
ii. Now, imagine pushing out one side of a polygon with n – 1 sides so that an n-sided polygon is
formed. Tell how many additional diagonals can be drawn. Then write a recursion equation for dn .