unit 5
in a golf tournament, the 16 golfers with the lowest scores win cash prizes. first place receive a cash prize of $1000, second place gets $950, third place gets $900,... find the sum of the prizes.
$10,000
the value of a car depreciates about 20% during its first year and about 10% each year after that. a car originally costs $25,999. what is the value of the car after 6 years of depreciation?
$12,281.72
jamie enjoys getting a consistent hourly raise of $0.75 per hour every 6 months of being an employee for Nike Outlet. he has worked at Nike Outlet for 2 1/2 years and just received his hourly raise. if he is currently making $18 per hour, what was his starting hourly wage?
$14.25
a small business sells $10,000 worth of skin care products during its first year. the owner of the business has set a goal of increasing annual sales by $7500 each year for the next 9 years. assuming that this goal is met, find the total sales during the first 10 years this business is in operation.
$437,500
find the sum of the first 100 terms of the series: 8.0+7.5+7.0+6.5+...
-1675
write the first four terms of the sequence 1. an = (-3)^(n) 2. a(k) = 8 + a(k-1), a1 = -5
-3,9,11,19 -5,3,11,19
find the sum of the first 8 terms of the series: 0.3+0.03+0.003+0.0003+...
0.333333 (repeated)
0! 1! 2! 3! 4!
1 1 2 6 24
writing terms of a sequence and graphing the sequence: find the designated term for each sequence: 1. a(subscript)n = ((-1)^n) / (2n - 1) a(subscript)4 = ? 2. a(subscript)n = 3 + (-1)^n a(subscript)2 = ? 3. list the first 4 terms of the sequence and graph it on the axis provided: a(subscript)n = 3n - 2
1. 1/7 2. 4 3. (1,1) (2,4) (3,7) (4,10) don't connect the points because this is a discrete graph (sequences you don't connect)
evaluate the expression involving factorials: 1. 11!/10! 2. 20!/(10!(9-1)!)
1. 11 2. 16,628,040
here is the formula for the nth term of a geometric sequence: an = 4*(.03)^(n-1) 1. write out the first 5 terms and find their sum 2. now write the series as a summation and verify your result with a graphing calculator 3. now find the sum of the infinite series
1. 4,0.12,0.0036,0.00108,0.00000324 sum = 4.12371124 2. 4.1271134 3. 4.12371134
for each sequence below, find the next three terms, the tenth term, the 50th term, and the sum of the first ten terms and the sum of the first fifty terms. 1. 1, 3, 5,... 2. 1, 4, 9,... 3. 32, 16, 8,...
1. 7, 9, 11; 19; 99; 100; 2500 2. 16, 25, 36; 100; 2500; 385; 42925 3. 4, 2, 1; 1/16; 5.68 x 10^-14; 63.94, 64
for each sequence below, find the next three terms, the 10th term, and the ratio of the tenth term to the ninth term. make a prediction for the ratio of the 50th term to the 49th term without performing a calculation. 1. 1, 1, 2, 3, 5,... 2. 3, -1, 2, 1, 3,...
1. 8, 13, 21; 55; 1.62; 1.62 2. 4, 7, 11; 29; 1.61; 1.62
find the common ratio between the following geometric sequences. find the closed formula for the nth term and use it to find a6. 1. 12, 36, 108, 324... 2. -1/3, 1/9, -1/27, 1/81...
1. a(subscript)n = 12 * 3^(n-1) a6 = 2916 2. a(subscript)n = (-1/3) * (-1/3)^(n-1) a6 = 0.00137
find the common difference for each of the following arithmetic sequences. find the closed formula for the nth term and use it to find a8. 1. 7, 11, 15, 19... 2. 2, -3, -8, -13...
1. a(subscript)n = 7 + 4(n-1) a8 = 35 2. a(subscript)n = 2 - 5(n-1) a8 = -33
recursive sequences: example 1: fibonacci sequence 1, 1, 2, 3, 5, 8 what is the pattern? how would this be written using a(subscript)k notation? example 2: a1 = 3 a(subscript)(k+1) = a(subscript)(k) - 2 find the next three terms in the sequence. example 3: a1 = -3 a(subscript)(k) = 2a(subscript)(k-1) find the next three terms in the sequence
1. add 2 previous terms together; a(subscript)(k) = a(subscript)(k-1) + a(subscript)(k-2) a1 = 1 a2 = 1 2. 3, 1, -1, -3 3. -3, 6, 12, 24
use finite difference to determine the degree of the sequence (function): 1. 4, 9, 14, 19, 24 2. 5, 8, 12, 17, 23 3. -7, 0, 19, 56, 117
1. linear (bc of the first order finite difference) 2. quadratic (bc of the second order finite difference) 3. cubic (bc of the third order finite difference)
evaluate: 15!/16! 6!/(3!(9-4)!)
1/16 1
find the sum of the first 15 terms: an = 3 * (3/4)^(n-1)
11.83963847
4. find the next three terms in the recursive sequence: a1 = 3 and a2 = 9 a(subscript)k = a(subscript)(k-1) + a(subscript)(k-2)
12, 21, 33
find the infinite sum: 100+20+4+0.8+...
125
find the sum of the first 5 terms: 5+15+25...
125
find the sum of the first 8 terms 3+7+11+15+...
136
find the sum of the first 5 terms: an = 300 * (1.06)^(n-1)
1391.127888
find the sum of the infinite series: 8+4+2+1+...
16
to add up add of the terms in a finite arithmetic sequence, the following strategy can be used: given Sn = 2+5+8+11+...an, find the sum of the first 10 terms. this can also be called the 10th partial sum because it's a sum of part of the sequence. if there are a reasonable number of terms, we can find the sum by listing the numbers in pairs.
2 29 5 26 8 23 11 20 14 17 by listing the number in pairs, we can add the pairs up and multiply the sum of pairs by the number of pairs. in this case we have 10 terms, so we have 5 pairs. each pair adds up to 31, and 5 * 31 = 155
generate the first 3 terms of the sequence 1. a(subscript)n = 2 + 7(n-1) 2. a(subscript)n = -3(n-1) 3. a(subscript)k = a(subscript)(k-1) + 8, a1 = -7
2, 9, 16 0, -3, -6 -7, 1, 9
write the first 3 terms of each sequence: 2. a(subscript)n = (-1)^(n-1) * (2/n) 3. a(subscript)n = {(n if n is even) (1/n if n is odd)}
2. 2, -1, 2/3 3. 1, 2, 1/3
find the sum of the first 10 terms: an = 2 * .25^(n-1)
2.666664124
find the sum of the positive integers from 205 up to 300
24,240
find the sum of the first 8 terms: an = (4/3)^(n-1)
26.96616369
find the sum of the first 6 terms of the series: 1+3+9+27+...
364
terms of sequences involving factorials: find the first three terms of the following sequence: a(subscript)n = (2^2) / n!
4, 2, 2/3
write the first 4 terms of the sequence and graph the sequence: a(subscript)n = (n + 1)^2
4, 9, 16, 25 (1,4) (2,9) (3,16) (4,25)
what is the sum of the first 20 odd integers 1+3+5+7+...+39
400
what is the sum of the first 100 positive integers 1+2+3+4+...+100
5050
sum key on calculator
MATH [0] summation SIGMA( x = 1 or 0 (on bottom) # of terms being added (on top) closed formula (in parentheses)
find the 150th partial sum of the arithmetic sequence 5,16,27,38,49,...
S150 = 123,675
finite arithmetic series formula:
Sn = (n/2)*(a + an)
sum of a finite geometric series:
Sn = a1 * (1-r^n) / (1-r)
sum of an infinite geometric series (when IrI is less than 1)
Sn = a1 / (1-r)
recursive form of a sequence
a formula in which each term is defined as a function of the proceeding term or terms
closed (explicit) form of a sequence definition
a formula that allows direct computation of any term in the sequence
sequence definition
a function whose domain is the set of positive integers. a sequences is usually represented by listing its values in order
arithmetic sequence defintion
a sequence generated by adding a constant difference to the previous term (linear)
geometric sequence definition
a sequence generated by multiplying the previous term by a common ratio (exponential)
infinite sequences definition
a sequence that never ends, usually indicated by.... after the last listed term
finite sequence
a sequence that only has a fixed number of terms
find the recursive formula of the arithmetic sequence whose 4th term is 28 and whose 8th term is 52. then list the 1st-5th terms.
a(k) = a(k-1) + 6 a1 = 10 10, 16, 22, 28, 34
find the recursive formula for the geometric sequence whose 1st term is 3 and whose common ratio is r = 2. then find a5
a(subscript)k = a(subscript)(k-1) * 2, a1 = 3 a5 = 48
find the recursive formula for the arithmetic sequence whose 2nd term is 8 and whose 12th therm is 28
a(subscript)k = a(subscript)(k-1) + 2
write a recursive formula to represent the following sequence which behaves similarly to the Fibonacci Sequence: 6, 10, 16, 26...
a(subscript)k = a(subscript)(k-1) + a(subscript)(k-2), a(subscript)k = 6, a(subscript)(k-1) = 10
recursive formula of an arithmetic sequence
a(subscript)k = a(subscript)(k-1) + d where a1 = ? **fill in values for a1 and d when writing the formula
recursive sequences: when writing recursive sequences, sometimes...
a(subscript)k notation may be used
find the nth (closed formula) for the arithmetic sequence beginning with -9 and then -6.
a(subscript)n = -9 + 3(n-1)
write the closed formula for the following sequence: 144, 72, 36, 18...
a(subscript)n = 144 * (1/2)^(n-1)
find the closed formula for the geometric sequence whose 1st term is 20 and whose common ratio is r = 1.05. then find a5
a(subscript)n = 20 * 1.05^(n-1) a5 = 24.31
find the nth term of the geometric sequence whose second term is 30 and fourth term is 67.5
a(subscript)n = 20 * 1.5^(n-1)
write the formulas for the given geometric sequence: 4, 8, 16, 32 closed: recursive:
a(subscript)n = 4 * 2^(n-1) a(subscript)k = a(subscript)(k-1) * 2, a1 = 4
write the formulas for the given arithmetic sequence: 4, -1, -6, -11 closed formula: recursive formula: use the closed formula above to find a15
a(subscript)n = 4 - 5(n-1) a(subscript)k = a(subscript)(k-1) - 5, a1 = 4 -66
find the closed formula for the arithmetic sequence whose 4th term is 20 and whose 13th term is 65. then find the 20th term.
a(subscript)n = 5 + 5(n-1) or a(subscript)n = 5n a20 = 100
recursive formula of a geometric sequence:
a(subscript)n = a(subscript)(k-1) * r, a1 = ? **fill in valves of a1 and r when writing the formula
closed formula of geometric sequence:
a(subscript)n = a1 * r^(n-1) **fill in valves for a1 and r when writing the formula
closed formula of an arithmetic sequence
a(subscript)n = a1 + d(n-1) **fill in valves for a1 and d when writing out the formula
let's look at the following geometric series: 200, 100, 50, 25,... what is a1? what is r?
a1 = 200 r = 1/2
find a2 and a5 in the following sequence: a(subscript)n = 1 / (n+1)!
a2 = 1/6 a5 = 1/720
an investment firm has a job opening with a salary of $30,000 for the first year. suppose that every year following, there is a 5% raise. find the compensation during your 40th year of work. complete 39 total raises since you don't get a raise in your first year).
a40 = $201,142.53
find the 5th term of the following geometric sequence: 4, 6, 9...
a5 = 20.25
depreciation: a tool and die company buys a machine for $135,000 and it depreciates at a rate of 30% per year. in other words, at the end of each year the depreciated value is only 70% of what it was at the beginning of the year. find the depreciated value of the machine after 5 full years. (complete 5 total depreciations)
a6 = $22,689.45
if the common ratio is greater than 1, we...
add larger and larger terms to our running sum. the total can get big in a hurry. using pairs doesn't do us much good in this case, because the pairs have different sums. fortunately, there is a shortcut formula for the sum of a finite geometric series that we will look at in a little bit.
the function f(x) = 3x - 2 has a domain of..., but the sequence a(subscript)n = 3n - 2 only has a domain of...
all real numbers; the set of positive integers
find the nth term of the geometric sequence whose 2nd term is 300 and whose 4th term is 18.75
an = 1200 * 0.25^(n-1)
write down the nth term of the given sequence. then generate the 6th term. 6, -10, -26, -42...
an = 6 - 16(n-1) a6 = -74
arithmetic series definition
an arithmetic series is the sum of consecutive terms in an arithmetic sequence defined as: S(subscript)n = a1 + a2 + .... + an where a1 is the first term and Sn is the sum of n terms ex: given the sequence 1,4,7,10,13,... the sum of the first terms is S6 = 1+4+7+10+13=51
a series is said to converge if it...
approaches a number other than positive or negative infinity for large value of n
why would using the closed formula be more efficient than the recursive formula?
because you can plug in and don't have to generate all terms
notice that the series below looks like an arithmetic sequence with on difference. what is the difference? Sn = 2+5+8+1+...an
can't add because not finitie S - add together a - looking for term
a series is said to diverge if it calls for us to...
continue adding larger and larger numbers because it will approach either positive or negative infinity.
look at problem 5-2: sequences and series with spreadsheets in packet: advanced precalculus - unit 5 sequences and series
cover check correct
when graphing the terms of sequence, the points should be...
discrete (not connected) because there are no values in between each term
a(subscript)n notation helps us...
distinguish between a function and a sequence
does the series from the last card converge or diverge
diverge bc r is greater than 1
if IrI is greater than or equal to 1, then the series...
does not have a sum
an infinite arithmetic series does NOT have a...
finite sum because it goes to infinity
because the sum starts at term 0 (n = 0 is below the summation sigma), the power...
is only n instead of n-1 **this would be a situation for like car deprecation**
find the common ratio and the sum of first 4 terms: an = (3^n) / 9
r = 3 13 1/3
the third term of a geometric sequence is 64 and the fifth term is 1024. find the 2nd term (assume that the terms of the sequence are positive).
r = 4 a2 = 16
diverge
r is greater than 1
converge
r is less than 1
d
the common difference between each successive term
r
the common ratio between each successive term
terms defintion
the ordered list of numbers in the sequence
factorials definition
the product of an integer and all the positive integers below it
geometric series definition
the sum of consecutive terms in a geometric sequence ex: 3+6+12+24+...an
a business did $12,000 in sales on Black Friday in 2016. the owner of the business has set a goal of increasing Black Friday sales by $1500 each year for 4 years. assuming that this goal is met, find the sales goals for the next 4 years ad find the total Black Friday sales from 2016-2020 (all 5 Black Fridays).
total = $75,000
learning about finite differences can help us determine...
what degree function might best define a sequence (note: this will only help us with linear, quadratic, cubic, etc.)
given the sequence 6, 10, 14, 18... determine whether 82 is a term in the arithmetic sequence. if so, what term?
yes 20th term
given the sequence 5, 15, 45... determine whether 3,645 is a term in the geometric sequence. if it is a term in the sequence, which term is it?
yes 7th term
example of finite arithmetic sequence and an infinite arithmetic sequence
{2, 3, 6, 8} - can add up 2, 4, 6, 8,... - can't add up
