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The people infected as the days go by: 8, 24, 72, 216, 648, … How many people will get infected in the 6th day? WHY???
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Sequences
A mysterious illness is affecting the residents of Mexico city.
Example 1: Epidemy
DAY NUMBER OF PEOPLE
INFECTEDFIRST DAY 8
SECOND DAY 24
THIRD DAY 72
The people infected as the days go by:
8, 24, 72, 216, 648, …
How many people will get infected in the 6th day?
WHY???
Example 2: Fibonacci numbers
Fibonacci considered the growth of an idealized (biologically unrealistic!!!) rabbit population
Newly born pair of rabbits: one male and one female
= 1 pair
One month later…
= 1 pair
Another month later…
= 2 pairs
A mating pair always produces one new pair (one male, one female) every month from the second month on…
How many pairs will there be in one year?
Rabbits never die!
Month 0: = 1 pair
Month 1: = 1 pair
Month 2: = 2 pairs
Month 3: = 3 pairs
Month 4: = 5 pairs
1, 1, 2, 3, 5, 8,13, 21, 34, …
What is the next number? WHY???
Fibonacci numbers:
Example 3: ProfitA company began doing business fouryears ago. Its profits for the last four
yearshave been $11 million, $15 million, $ 19million and $23 million.
If the PATTERN continues the expectedprofit in 30 years is going to be:
$127 millionWHY????
All of the above are…
SEQUENCES
A SEQUENCE is a set of quantities arranged in a definite order. For example:
1, 2, 3, 4, 5, … 1, 4, 9, 16, 25, … 1, 8, 27, 64, 125, … -10, -8, -6, -4, -2, …
But… what is a sequence?
TWO TYPES OF SEQUENCES
Arithmetic Sequence1, 3, 5, 7, 9, …
11, 15, 19, 23, …Geometric Sequence
2, 6, 18, 54, 162200, 20, 2, 0.2
How to distinguish an arithmetic series?
An arithmetic sequence will always have a
common difference between successiveterms. For example: 2, 4, 6, 8, 10, … COMMON DIFFERENCE of 2
1, 4, 7, 11, 14, … COMMON DIFFERENCE of 3
GETTING BACK TO THE PROFIT EXAMPLE …
How can you calculate the 30th term?
Moreover, how can you calculate the nth term?
Hint: What is the common difference?
YEARS PROFIT1 11 mm2 15 mm3 19 mm4 23 mm… …30 ????
YEARS PROFIT1 11 mm2 15 mm3 19 mm4 23 mm…30
GETTING BACK TO THE PROFIT EXAMPLE …
The common difference is 4.
15 – 11 = 419 – 15 = 4
Therefore we know that we need to multiply the nth
times 4
But … 4(1) = 4,
4(2) = 8 and 4(3) = 12 …
If we add + 7 we’ll get the result.
Hence, in 30 years the profit will be
30(4) + 7 =
YEARS PROFIT1 11 mm2 15 mm3 19 mm4 23 mm…30 127 mm
GETTING BACK TO THE PROFIT EXAMPLE …
HOW WOULD YOU CALCULATE
THE Nth TERM???
HOW WOULD YOU CALCULATE
THE Nth TERM??? HINT: Look for it in your booklet!!!!!
A geometric sequence will always have acommon ratio between successive terms.For example:
2, 4, 8, 16, 32, … COMMON RATIO of 2
3, -9, 27, -81, 243, … COMMON RATIO of -3
How to distinguish a geometric sequence?
A mysterious illness is affecting the residents of Mexico city.
Getting back to the epidemy example…
DAY NUMBER OF PEOPLE
INFECTEDFIRST DAY 8
SECOND DAY 24
THIRD DAY 72
Getting back to the epidemy example…
DAY NUMBER OF PEOPLE
INFECTEDFIRST DAY 8
SECOND DAY 24
THIRD DAY 72
… …
20th DAY ???
How can you calculate the 20th term?
Moreover, how can you calculate the nth term?
Hint: What is the common ratio?
Getting back to the epidemy example…
DAY NUMBER OF PEOPLE
INFECTEDFIRST DAY 8
SECOND DAY 24
THIRD DAY 72
… …
20th DAY 9,298,091,736
The common ratio is 3
8x3= 248x3x3 = 72
8x3x3x3 =216
Therefore we knowthat we need to multiply 8x3x3x…x3
19 times
HOW WOULD YOU CALCULATE
THE Nth TERM???
HOW WOULD YOU CALCULATE
THE Nth TERM??? HINT: Look for it in your booklet!!!!!
Your parents give you an allowance of 5 euros (!!!) a week when you are 5 years old. Every year your allowance is increased by 1 euro. You receive an allowance every week until your 21st birthday.
Your parents give you an allowance of 5 euros (!!!) a week when you are 5 years old. Every year your allowance is increased by 12%. You receive an allowance every week until your 21st birthday.
What would you choose???
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