Patterns and Sequences

You win a contest and are offered cash money for 14 days You win a contest and are offered cash money for 14 days. Which one of these would you choose? Why? 1st choice 2nd choice You receive $500 the first day, $550 the second day, $600 the third day, $650 the fourth day, etc. You continue to receive this for 14 days. You receive $1 the first day, $2 the second day, $4 the third day, $8 the fourth day, etc. You continue to receive this for 14 days.

You should have chosen 2nd choice The first method initially gives you more but after 14 days you have $12,750. The second method initially gives you less but would accumulate to $16, 383.

Arithmetic Sequences In the first example, the common difference was $50 each time. $500 to $550 to $600 to $650, etc. Since we are adding(you can also subtract in other examples), this example is deemed an arithmetic sequence.

Try to find the common difference in these 2 examples (1)8, 13, 18, 23, …… Common difference is +5 (2)12, 9, 6, 3, ………. Common difference is -3

Geometric Sequences In the second example your cash reward is multiplying from 1 to 2 to 4 to 8 so the common ratio is x 2 The common ratio can be either multiplying or dividing If you multiply or divide you are using a geometric sequence.

Try these Geometric Sequences and find the common ratio 4, 12, 36, 108, ……. Common ratio is x 3 (2) 4, 2, 1, 0.5, …….. Common ratio is ÷ 2

Is the following set of numbers a geometric or arithmetic sequence Is the following set of numbers a geometric or arithmetic sequence? Explain your answer. 4, 6, 9, 13, 18, ………..

NEITHER ARITHMETIC OR GEOMETRIC ANSWER NEITHER ARITHMETIC OR GEOMETRIC Although we could write more terms in the sequence, there is not a common difference or common ratio.

Finding the Type of Sequence TEACHER EXAMPLE Write the next three terms in the sequence and tell whether it is arithmetic, geometric, or neither. -1, 3, -9, 27, …………. Well to get from –1 to 3, I can add 4 or multiply by –3. But adding 4 to three will NOT give me 9, so I will try to multiply 3 by –3 and that equals to –9. The next three terms would be –81, 243, and –729 and it would be a geometric sequence since I have a common ratio

NOTEBOOK EXAMPLES Write the next three terms of each sequence and then tell whether it is arithmetic, geometric, or neither. 3, 9, 27, 81, …………… -12, 12, -12, 12, …………… 10, 13, 18, 25, ……………. 50, 200, 350, 500, …………..

ANSWERS 3, 9, 27, 81, 243, 729, 2,187 (Geometric) -12, 12, -12, 12, -12, 12, -12 (Geometric) 10, 13, 18, 25, 34, 45, 58 (Neither) 50, 200, 350, 500, 650, 800, 950 (Arithmetic)

THINKER #1 The population of the US was as follows: 1980- about 226 million 1985- about 238 million 1990- about 250 million 1995- about 263 million Would it be better to call this a geometric or arithmetic sequence? Either One since both the common ratio and common difference are about the same

THINKER #2 A runner training for a marathon starts by running 2 miles on the first day. They increase their distance by 1.5 miles. On what day of training will the marathon runner run more than the normal marathon distance of 26.2 miles? DAY 18