1. describe and extend each sequence. a. the height, in feet, of windowsills in a high-rise building is represented by this sequence: 3, 15, 27, 39, describe the rule for the sequence. find the next two terms of the sequence. show your work. b. mason put sticky notes on some pages of his math book. the page numbers form this sequence: 3, 5, 8, 13, 21, 34, describe the rule for the sequence. find the next two terms of the sequence. show your work. the explicit formula 2. use the following arithmetic sequence and the formula an = a1 + (n – 1)d to answer the questions below. 123, 116, 109, 102, 95, part i: find the value of each of the following: a1 = and d = part ii: find the explicit formula. show your work. part iii: use the explicit formula you found in part ii to find the value of the 100th term in the sequence, a100. show your work. 3. state whether each sequence is arithmetic or geometric, and then find the explicit and recursive formulas for each sequence. (10 points: 5 points each) formulas: part i: the total value of a collection of nickels can be described by this sequence: 5, 10, 15, 20, 25, 30, type of sequence: explicit formula: recursive formula: part ii: the number of rats living in an abandoned building increases each year. the total number of rats can be described by this sequence: 2, 6, 18, 54, 162, type of sequence: explicit formula: recursive formula: 4. a stone nudged off the royal gorge bridge near cañon city, colorado, falls 1053 feet before hitting water. because its speed increases as it falls, the distance it travels each second increases. during the first second, it drops 16 feet. during the next second, it drops an additional 48 feet. during the third second, it drops another 80 feet. the distances traveled each second form an arithmetic sequence: 16, 48, 80, part i: how far does the stone fall during the 5th second? find and use the explicit formula. what is the first term of the sequence? what is d, the common difference? write the explicit formula in function notation. use f(n) = f(1) + (n – 1)d, where f(1) represents the first term. use the explicit formula to find the distance the stone travels in the 5th second. part ii: the table below shows the values in the sequence you already know. use the explicit formula or the common difference to complete the table for the first 7 seconds. time (s) 1 2 3 4 5 6 7 distance (ft) 16 48 80 144 part iii: use the table from part ii to answer the questions. the values in the table form a(n) sequence. the term values are shown in the row, and the term numbers are shown in the row. this sequence is associated with a(n) function. the domain of the function is the set of time values:

1A: To get from 3 to 15, you add 12. To get from 15 to 27, you add 12. To get from 27 to 39, you add 12...
...so that's the rule; add 12.
As an explicit function it would be f(x) = 12x - 9 (where y is the number itself and x is where the number is in the sequence--the first number is x = 1, the second, x = 2, and so on). This is because the sequence increases by 12 each time, and it starts at 3 (and 12 - 9 = 3).

1B: 3+5=8, 5+8=13, 8+13=21, 13+21=34.
The rule for this sequence (which is actually a famous sequence, commonly found in nature, callex Fibonnacis sequence) is that you add the previous number in the sequence to the current number in the sequence to get to the next number in the sequence.

2: an = a1 + (n-1)d is used to find any number in a sequence using n, or where the number is in the sequence.
(If you wanted to find the 5th number, for example, n would equal 5.)
Therefore, an is the nth number in the sequence and a1 is the first number in the sequence.
d is the common difference (the difference between any two consecutive terms in the sequence that is the same throughout the whole sequence).
(In question 1A, the common difference is 12.)

Therefore, for the sequence 123, 116, 109, 102, 95, ...

Part I: a1 is 123, and d is -7, because 116 minus 123 is negative 7 (and the sequence is decreasing).

Part II: The explicit formula is y = -7x + 130. Using our our equation an = a1 + (n-1)d and substituting in our values, we get...
y = 123 + (x-1)-7 (n = x and an = y)
= 123 + -7x + 7 (Distribute)
= -7x + 130 (Simplify)

Part III: We just plug in 100 for our x value...
f(x) = -7(100) + 130
= -700 + 130
= -570
The 100th term is -570.

3. An arithmetic sequence translates to a linear function; geometric becomes exponential, quadratic, cubic, etc.
As you know, arithmetic sequences have a common difference. That means the difference between any two consecutive terms is the same.
Geometric sequences have a common ratio. That means the quotient between any two consecutive terms is the same.

Part I: 5, 10, 15, 20, 25, 30, ...
Type of Sequence: Arithmetic
(It has a common difference--5)
Explicit Function: y = 5x
an = a1 + (n-1)d
y = 5 + (x-1)5
= 5 + 5x - 5
= 5x
Recursive Function: next = now + 5
The next term equals this term plus 5.

Part II: 2, 6, 18, 54, 162, ...
Type of Sequence: Geometric
(It has a common ratio--3)
Explicit Function: y = 2/3 * 3^x
Geometric functions have a different equation. r is the common ratio.
an = a1 * r^n-1
y = 2 * 3^x-1
= 2 * 3^x / 3
= 2/3 * 3^x
Recursive Function: next = now * 3, starting at 2
The next term equals this term times 3, starting at the number 2.

4: 16, 48, 80, ...

Part I:
The first term: 16
The common difference: 32
an = a1 + (n-1)d
y = 16 + (x-1)32
y = 16 + 32x - 32
y = 32x - 16
That's our explicit function. To find the 5th term, we substitute in 5 for x...
y = 32(5) - 16
y = 160 - 16
y = 144

Part II:
Second: 1 2 3 4 5 6 7
Distance: 16 48 80 112 144 176 208
y = 32x - 16
y = 32(4) - 16 = 128 - 16 = 112
y = 32(6) - 16 = 192 - 16 = 176
y = 32(7) - 16 = 224 - 16 = 208

Part III:
The values in the table form a(n) (arithmetic) sequence. The term values are shown in the (second) row, and the term numbers are shown in the (first) row. This sequence is associated with a(n) (linear) function. The domain of the function is the set of time values: (1, 2, 3, 4, 5, 6, 7)

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