Download Slide 1 - Mrs. Hille`s FunZone

Look for a
Pattern
Example
Since his first birthday, André’s
uncle has given him a gift of
money such that the number of
dollars matched his age. Today
is André’s fourteenth birthday.
How much money has he been
given over the years by his
uncle?
Example
If André’s uncle continues
giving him these gifts until
Andre is 50, how much money
will he have received by then?
Think: Determine what you are
trying to find.
You want to know the total
amount of money Andre
received from this uncle in
14 years and how much he
will have received after 50
years.
Think: How could you find this
sum? Is there an easier way
than just adding the numbers
from 1 to 14 and then adding
the numbers from 1 to 50? Is
there a pattern to this sum? Is
there a rule based on the year
number that can be used to
find the sum in dollars?
Look at the first few years to
try to determine a pattern in
the sums. Making a table
may help you organize the
information.
#
Sum in Dollars
1
2
3
4
5
6
7
1
1+2=3
1+2+3=6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
There definitely is a pattern
in the sums. The next
number in the sequence can
be found in the previous
number.
Think: How can you find the
sum of the numbers to 14 and
50 without writing down the
entire list of numbers?
Notice that the numbers that
are added to get 28 in the
table can be added in pairs
that always give the same
numerical result. That is, 1 +
7 = 8, 2 + 6 = 8, and 3 + 5 = 8.
Since there are 7 numbers in
the sum, there are 3 pairs of
numbers and an unpaired
middle number. The middle
one is one half the sum of
one of the pairs.
Think: What will be the sum of
the first and last terms in
1 + 2 + 3 + … + 13 + 14? How
many pairs will there be in the
sum?
1 + 14 = 15, as does 2 + 13.
There are 7 pairs with no
remainder.
14 years: 7 x 15 = $105
Another way to look at this
same problem is to look at
the sum of two sets of
numbers from 1 to 7. The
answer is obviously twice
what is needed and requires
you to divide by 2.
sum
1
2 3 4 5 6 7
7
6 5 4 3 2 1
8
8 8 8 8 8 8
1+2+3+4+5+6+7
7(7 + 1)
=
2
7x8
=
2
= 28
Likewise, considering the
sum of two sets of numbers
from 1 to 50, there is twice
what is needed in the
product of (1 + 50) and 50.
Thus the sum
1 + 2 + 3 + 4 + 5 + … + 50 =
50(1 + 50) = 1,275
2
Think: How can this be
generalized for a sum of the
first n whole numbers?
Following this same pattern,
the sum of the first n
whole numbers,
1 + 2 + 3 + 4 + 5 + … + n,
can be found.
Find the common sum
(n + 1), multiply it by n, and,
realizing that this is twice the
needed sum, divide by 2 to
get the correct answer.
1+2+3+4+5+…+n
n(n + 1)
=
2
Now try this out with some of
your examples to convince
yourself that it works.
1+2+3+4+5+6
6(6 + 1)
=
2
= 21
This answer matches the
value of the sum in the first
six whole numbers in the
table.
Exercise
Square numbers are numbers
that can be displayed by a
pattern of dots that form a
square. Give the number of
dots in each square shown.
Exercise
Draw the next square; then
give the rule for finding the
nth term of this sequence.
Exercise
Look at the pattern of sums
given below. Write the next
four patterns and their sums.
1=1
1+3=4
1+3+5=9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
Write a rule for finding the
sum based on the quantity of
odd numbers n in the sum.
Find the sum of the first 20
odd numbers by using your
rule. Then use a calculator to
check your sum by adding
the first 20 odd numbers
together.
Exercise
An eccentric math teacher
told his class that he would
assign one problem on the
first day of school, two
problems on the second day,
four problems on the third
day, and so on.
At this rate, how many
problems would he assign on
the tenth day? Find a rule
that can be used to determine
the number of problems
assigned on the nth day.
Exercise
Triangular numbers are
numbers that can be
displayed by a pattern of dots
that form a triangle. The first
four triangular numbers are
1, 3, 6, and 10.
What are the next four
triangular numbers? What is
the rule for finding the value
of a triangular number based
on the number of the term in
the sequence?
Exercise
Find the sum of the first eight
positive multiples of 11. Write
out this sum. If you add the
first and last addends, then
the second and next to last,
and continue in this pattern,
what is the common sum of
each pair?
How many pairs of addends
are there in this sum? How
does the number of pairs
relate to the number of
addends? What could you do
with the common sum and
the number of pairs to find
the entire sum?
Exercise
Find the sum of the first ten
natural numbers and then the
first one hundred natural
numbers. Give a rule that can
be used to find the sum of
the first n natural numbers.
Exercise
In the last two problems, the
number of addends was
always even. Verify that this
same rule will work for
finding the sum of the first
five natural numbers. Do you
think this rule works for all
natural numbers?
Exercise
The following pattern is called
the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, …
Describe how to find the next
term in the pattern using the
terms that already exist. Find
the next three terms of the
sequence.