Vocabulary
The fixed number is a Geometric Sequence.
What is the common ratio
The value of an ounce of silver is about $16 and over the last several years silver has increased in value by about 7%. How much should an ounce of silver be worth 20 years from now?
16, 17.12, 18.32, 19.60, 20.94, ...
a20= (17.12)(1.05)20.1
= $61.91
Determine whether the sequence is an arithmetic sequence. If so, find the common difference and the next three terms.
-10, -7, -4, -1, ...
Yes
-10, -7, -4, -1, 2, 5, 8
Find the next few terms of the Geometric Sequence
3, 6, 12, 24, ...
Then find a24.
3, 6, 12, 24, 48, 96, 196.
a24= 5(2)23
a24= 25,165,824
Who discovered this technique of math?
Who is Carl Friedrich Gauss
Part of a sequence that refers to the algebraic representation of any given term in a sequence.
What is a nth term.
A person has having a graduation party and noticed that only 5 people were there after the first hour but grew in size by 61% every hour. If the size of the party grew this way for 6 hours, how many people would be at the party on the 6th hour?
5, 8, 13, 21, 34, 54
an= 5(1.61)n-1
=54
Find the S62 of the following series: 4+9+14+19+...
n=62 a1=4 an=?
an=a1+(n-1)d
a62= 4+(n-1)d
a62= 4+(62-1)5
a62=309
S62=n/2 (a1+an)
S62= 62/2 (4+309)
S62= 31(313)
S62= 9,703
Determine the fraction for the following repeating decimal using Infinite Geometric Series formulas.
0.99999999
10x= 9.9999999
x=0.99999999
9x=9/9
x=1
How old was Carl Friedrich Gauss when he discovered this new math Technique?
8 years old
Used to find the nth term of a sequence. A formula that allows direct computation of any term for a sequence a1, a2, a3, ..., an, ....
What is the Explicit Formula.
Kelly starts to save money. On the first week of the year, she saves one cent ($0.01). Then, for each week that follows she continues to double double the amount she saved the previous week. So, on the second week she saved an additional 2 cents ($0.02) and the 3rd 4 cents ($0.04). If this process were able to be continued for the entire year of 52 weeks, how much money would Kelly have saved by the end of the year?
Sn= a1(1-rn)/1-r
=0.01(1-252)/1-2
=4.5036x1015
=$45,036,000,000,000
Given that a sequence is arithmetic, a1=5, and the common difference is 4, find a37.
an= a1+(n-1)d
a37=5+(37-1)(4)
a37= 5(36)(4)
a37=149
Determine the fraction for the following repeating decimals using the Infinite Geometric Series Formula.
14.14141414
100x= 1414.141414
x= 14.14141414
99x=1400/99
x= 1400/99
What math technique did Carl Friedrich Gauss discover?
Finding the partial sum of any Arithmetic Series.
A sequence of terms that have a common difference between them.
What is a Arithmetic Sequence
Jeff want to invest $4,000 in a retirement fund that guarantees a return of 8% annually using continuously compound interest. How many years and months will it take for his investment to double?
Continuous Compound Interest: P=P0ert
8,000= 4,000e0.08t
2=e0.08t
ln(2)=ln(e0.08t)
ln(2)=0.08t
8.67=t
Given the terms 2, 4, 6, 8,..., Find the next three terms as well as find a42. Determine the Recursive Definition and Explicit Definition.
2, 4, 6, 8, 10, 12, 14
a=2
42 (2)= 84
a42=84
Recursive Definition: an= an-1+2
Explicit Definition: an=2n
72+36+18+9+4.5+... ... ...=
r=1/2 a1=72
S5= (72(1-(0.5)5))/1-0.5
= 139.5
S10= 143.8...
S50=144
S90= 144
S∞ = 144
How did Gauss find the partial sum of any Arithmetic Series?
He determined that if you find the sum of the most outer pair of numbers its sums to 101 and that the next inner paor after that sums to 101 and so on. There should be 50 pairs of numbers that sums to 101.
A sequence of numbers in which the ratio between any two consecutive terms is the same. In other words, you multiply by the same number each time to get the next term in the sequence.
What is a Geometric Sequence
Jim put his $15,000 into a high yields savings account that pays 2.8% annually. The account is compounded annually. If the bank uses a compound interest formula, how much will the account be worth in 5 years if left untouched?
Compound interest formula: P= P0 (1+ r/n)nt
=1500 (1+ 0.028/1)(1*5)
=1500(1.028)5
= $17,220.94
Determine the Sum of the following partial arithmetic series using the formula.
30+26+22+... ... ...+ (-102) + (-106)=
n=? a1=30 an=-106
an= a1+(n-1)d
-106=30+(n-1)d
-106= 30+(n-1)(-4)
-106= 30-4n+4
-140=-4n
35=n
S35= 35/2 (30+106)
S35= 17.5 (-76)
S35= -1330
1/3+ 1/6+ 1/12+ 1/24+ 1/48+ ... ... ..=
a1= 1/3 r=1/2
S= a1/1-r
=(1/3) / (1/2)
= 1/3 * 2/1
= 2/3
What is the formula Gauss discovered to find the partial sum of any Arithmetic Series?
What is Sn= n/2 (a1+an)