The objective for this section is to find and create geometric sequences.
The ratio between consecutive terms in geometric sequences is the constant. Which is also known as the "Common Ratio". The common ratio= a2 divided by a1; a3 divided by a2; and so on. Some examples are: 5,15,45,135... 15/5=3 45/15=3 135/45=3 so it is geometric and the common ratio is 3. **BUT be CAREFUL because although some sequences may have a pattern, they may not neccisarily be geometric** 15,30,45,60... 30/15=2 45/30=1 1/2 The common ratios are not that common now are they? Therefore, this sequence is not geometric.
Geometric Sequence Formulas Recursive: An=An-1(r) Explicit: An=(A1)r^n-1 Where An is the "nth term". A1 is the first term. "n" is the order/position of the term. and r is the common ratio.
EXAMPLES!!! yay!! -Find the 19th term in the sequence- a) 11,33,99,297... A19=11(3^18) A19= 4261625379 b) 20,17,14,11,8... A19= 20(.85^18) A19=1.073
GRAPHS (Although I cannot graph on here, I will walk you through the steps.) If you have numbers such as 2,4,6,8... your coordinates would be (1,2) (2,4) (3,6) and (4,8) because 2 is your 1st term, 4 is your 2nd term and so forth. This is an arithmetic sequence and the graph will be linear. Now if you have numbers like 1,3,9 your coordinates will work the same, but you will have an exponential graph because it is a geometric sequence.
GeOmEtRic MEANS To find the geometric mean of any 2 positive numbers you simply take the positive square root of the product of the 2 numbers. such as 3, ,3 you take the square root of 9 and your middle term (your mean) is 3. **DON'T FORGET** take your + or -. If the 2 outside terms are negative then the middle is negative and the terms around it are both positive. ex. -a, , , ,-e -a,+b,-c,+d ,-e
I teach AP Calculus, Pre AP Precalculus, and Advanced Algebra/Trigonometry at Muscle Shoals High School. In my spare time, I enjoy spending time with my family and granddaughters. My husband and I like to travel and visited Germany and France last summer. I also enjoy reading, art and music.
Last spring, I read The Other Boleyn Girl by Philippa Gregory. I am currently reading the Constant Princess (by the same author).
I love "anything about England".
"SUGGESTED READING LIST" The Mathematical Traveler. It is a great book that combines history and interesting facts about famous mathematicians. See me if you want to "check it out".
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GEOMETRIC SEQUENCES =)
The objective for this section is to find and create geometric sequences.
The ratio between consecutive terms in geometric sequences is the constant. Which is also known as the "Common Ratio". The common ratio= a2 divided by a1; a3 divided by a2; and so on.
Some examples are:
5,15,45,135...
15/5=3 45/15=3 135/45=3
so it is geometric and the common ratio is 3.
**BUT be CAREFUL because although some sequences may have a pattern, they may not neccisarily be geometric**
15,30,45,60...
30/15=2 45/30=1 1/2
The common ratios are not that common now are they? Therefore, this sequence is not geometric.
Geometric Sequence Formulas
Recursive: An=An-1(r)
Explicit: An=(A1)r^n-1
Where An is the "nth term". A1 is the first term. "n" is the order/position of the term. and r is the common ratio.
EXAMPLES!!! yay!!
-Find the 19th term in the sequence-
a) 11,33,99,297...
A19=11(3^18)
A19= 4261625379
b) 20,17,14,11,8...
A19= 20(.85^18)
A19=1.073
GRAPHS
(Although I cannot graph on here, I will walk you through the steps.)
If you have numbers such as 2,4,6,8... your coordinates would be (1,2) (2,4) (3,6) and (4,8) because 2 is your 1st term, 4 is your 2nd term and so forth. This is an arithmetic sequence and the graph will be linear.
Now if you have numbers like 1,3,9 your coordinates will work the same, but you will have an exponential graph because it is a geometric sequence.
GeOmEtRic MEANS
To find the geometric mean of any 2 positive numbers you simply take the positive square root of the product of the 2 numbers.
such as 3, ,3 you take the square root of 9 and your middle term (your mean) is 3.
**DON'T FORGET**
take your + or -. If the 2 outside terms are negative then the middle is negative and the terms around it are both positive.
ex.
-a, , , ,-e
-a,+b,-c,+d ,-e
GOOD LUCK!!
and have a WONDERFUL day!! =)
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