ionGeometric Progression, Series & Sums
Introduction
A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where| r| common ratio|
| a1| first term|
| a2| second term|
| a3| third term|
| an-1| the term before the n th term|
| an| the n th term|
The geometric sequence is sometimes called the geometric progression or GP, for short. For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3. The geometric sequence has its sequence formation:

To find the nth term of a geometric sequence we use the formula:

where| r| common ratio|
| a1| first term|
| an-1| the term before the n th term|
| n| number of terms|

Sum of Terms in a Geometric Progression
Finding the sum of terms in a geometric progression is easily obtained by applying the formulas: nth partial sum of a geometric sequence

sum to infinity

where| Sn| sum of GP with n terms|
| S∞| sum of GP with infinitely many terms|
| a1| the first term|
| r| common ratio|
| n| number of terms|

Examples of Common Problems to Solve
Write down a specific term in a Geometric Progression
Question
Write down the 8th term in the Geometric Progression 1, 3, 9, ... Answer

Finding the number of terms in a Geometric Progression
Question
Find the number of terms in the geometric progression 6, 12, 24, ..., 1536 Answer

Finding the sum of a Geometric Series
Question
Find the sum of each of the geometric series| |
Answer

Finding the sum of a Geometric Series to Infinity
Question

Answer

Converting a Recurring Decimal to a Fraction
Decimals that occurs in repetition infinitely or...

...1. What is the sum of the geometric sequence 8, –16, 32 … if there are 15 terms? (1 point)
= 8 [(-2)^15 -1] / [(-2)-1]
= 87384
2. What is the sum of the geometric sequence 4, 12, 36 … if there are 9 terms? (1 point)
= 4(3^9 - 1)/(3 - 1)
= 39364
3. What is the sum of a 6-term geometric sequence if the first term is 11, the last term is –11,264 and the common ratio is –4? (1 point)
= -11 (1-(-4^n))/(1-(-4))
= 11(1-(-11264/11))/(1-(-4))
= 2255
4. What is the sum of an 8-term geometric sequence if the first term is 10 and the last term is
781,250? (1 point)
=8 (1-390625)/(1-5)
=781,248
For problems 5 8, determine whether the problem should be solved using the formula for an arithmetic sequence, arithmetic series, geometric sequence, or geometricseries. Explain your answer in complete sentences. You do not need to solve.
5. Jackie deposited $5 into a checking account in February. For each month following, the deposit
amount was doubled. How much money was deposited in the checking account in the month of August? (1 point)
To solve this, a geometric sequence is used because the terms share a constant ratio as 2.
6. A local grocery store stacks the soup cans in such a way that each row has 2 fewer cans than
the row below it. If there are 32 cans on the...

...… is a quadratic sequence.
2.1
Write down the next term.
(1)
2.2
Determine an expression for the term of the sequence.
(4)
2.3
What is the value of the first term of the sequence that is greater than 269?
(4)
[9]
QUESTION 3
3.1
The first two terms of an infinite geometric sequence are 8 and . Prove, without the use of a calculator, that the sum of the series to infinity is .
(4)
3.2
The following geometricseries is given: x = 5 + 15 + 45 + … to 20 terms.
3.2.1
Write the series in sigma notation.
(2)
3.2.2
Calculate the value of x.
(3)
[9]
QUESTION 4
4.1
The sum to n terms of a sequence of numbers is given as:
4.1.1
Calculate the sum to 23 terms of the sequence.
(2)
4.1.2
Hence calculate the 23rd term of the sequence.
(3)
4.2
The first two terms of a geometric sequence and an arithmetic sequence are the same. The first term is 12. The sum of the first three terms of the geometric sequence is 3 more than the sum of the first three terms of the arithmetic sequence.
Determine TWO possible values for the common ratio, r, of the geometric sequence.
(6)
[11]
QUESTION 5
Consider the function
5.1
Write down the equations of the...

...CHAPTER 7 ARITHMETIC AND GEOMETRICPROGRESSIONS
7.1 Arithmetic Progression (A.P)
7.1.1 Definition
The nth term of an arithmetic progression is given by
,
where a is the first term and d the common difference. The nth term is also known as the general term, as it is a function of n.
7.1.2 The General Term (common difference)
Example 7-1
In the following arithmetic progressions
a. 2, 5, 8, 11, ...
b. 10, 8, 6, 4, ...
Write (i) the first term, (ii) the common difference,
(iii) (iv)
Solution i) a. 2 b. 10
ii) a. 5-2 = 3 b. 10 -8 = 2
iii) a. = 14
b. = 2
iv) a. = 59
b. = 48
Example 7-2
How many terms are there in the following arithmetic progression?
(i) 3, 7, 11, 15, ... , 79. (ii)
Solution i)
ii)
7.1.3 The sum of the First n-Terms
or
Example 7-3
a. Find the sum of 20 terms of the following arithmetic progression:
(i) 1 + 2 + 3 + 4 + ... (ii) 5 + 1 + (3) + (7) + ...
Solution (i)
(ii)
b. Find the sum of the following arithmetic progression:
(i) 13 + 17 + 21 + ... + 49. (ii) 2.3 + 2.7 + 3.1 + ... + 9.9.
Solution (i)
(ii)
7.1.4 Solving Questions on Arithmetic Progression
The questions on arithmetic progression usually involve
(i) a term and another term,
(ii) a term and a...

...Assessment 09.08 GeometricSeries Activity
Material list:
three different balls of various sizes and textures
measuring tape or yardstick
a blank wall
a step stool or chair
a family member or friend
Procedures:
1. Choose a height from which all of the balls will be dropped one at a time.
2. Vertically along the blank wall, set up the measuring tape and step stool or chair.
3. Have a family member or friend stand on a step stool and drop one of the balls from the chosen height away from the measuring tape.
4. Face the measuring tape, opposite the ball’s starting point from about 7 or 8 feet. As the ball falls, measure the height of the ball on four consecutive bounces. (You may need to repeat the process to ensure that your measurements are accurate. You may choose to video each drop to assure accuracy.)
5. Write the height of each bounce, beginning with the height from which the ball originally fell,
in the chart below.
Ball 1
Description:
Ball 2
Description:
Ball 3
Description:
Height 1
(starting point)
Height 2
Height 3
Height 4
Height 5
6. Repeat the process with each ball. Be sure that each ball is originally dropped from the same height.
7. Beginning with Height 1, plot the height number (1, 2, 3, …) on the x-axis and the corresponding height measurement on the y-axis in GeoGebra. You may do this by using the New Point icon in the toolbar or by typing each ordered pair in the Input bar.
Be sure...

...What is the closed-form expression for the below sum of GeometricProgression (GP) sequence, S n ?
S n a aR aR 2 ... aR n
(1)
where R is called the common ratio (between consecutive terms) of the GP
sequence.
The reason why we want to derive a closed-form expression for S n is for the sake
of calculating the summation, or otherwise we need to add all terms one-by-one
together, which does not make a sense if the number of terms is huge, say a
million terms!
Most importantly, we based on the closed-form expression to derive the PV
and FV expressions for both ordinary annuity and annuity due.
Steps:
1. Multiply the both sides of equation (1) by the common ratio, R , to have
S n R aR aR 2 aR 3 ... aR n1 (2)
2. Then subtract equation (1) by equation (2), (or vice versa; it doesn’t matter
which subtracts which as the result will be the same.), i.e.,
S n S n R (a aR aR 2 ... aR n ) aR aR 2 aR 3 ... aR n1 (3)
3. Notice that all terms on the right hand side except for the first and last term,
a, aR n1 , are cancelled. So, equation (3) becomes,
(4)
S n (1 R) a aR n1
4. Remember our objective is to calculate S n . From equation (4), S n is obvious to
equal to,
a aR n 1 a(1 R n1 )
Sn
(1 R)
(1 R)
(*)
5. We are done. Equation (*) is the closed-form expression of that we want to
obtain.
Applications:
1. The PV for ordinary annuity, e.g....

...Arithmetic Progressions (AP)
An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms in an arithmetic progression are usually denoted as T1 , T2 , and T3 , where T1 is the initial term in the progression, T2 is the second term, and so on. Thus, Tn is the nth term of the arithmetic progression. An example of an arithmetic progression is….
2; 4; 6; 8; 10; 12; 14;
Since the difference between successive terms is constant, we have….
T3 - T2 = T2 – T1
Thus, in general, we will denote the difference of the two consecutive arithmetic progression terms as “d”, which is a common notation.
GeometricProgressions (GP)
A geometricprogression is a list of terms as in an arithmetic progression but in this case the ratio of successive terms is a constant. In other words, each term is a constant times the term that immediately precedes it. Let us write the terms in a geometricprogression as T1 , T2 , T3 and so on. An example of a geometricprogression is
10; 100; 1000; 10000;
Since the ratio of successive terms is constant, we have….
Thus, the ratio of successive terms is usually denoted by r and the first term again is usually written as T1. If we know the first term in a...

...pair-wise to form a closed chain. Regular polygon is a polygon that is equiangular and equilateral.
9. 3- triangle 4- Quadrilateral 5-Pentagon 6-Hexagon 7-Heptagon 8-Octagon
9-Nonagon 10-Decagon 11- hendecagon 12-dodecagon 13-tridecagon 14-tetradecogon
15-pentadecogon 10000-myriagon
10. An Interior Angle is an angle inside a shape. (n-2) × 180° number of sides minus 2 times 180 equals interior angles.
11. LEONARDO DA VINCI, 'The Last Supper' 1494-98, Perspective Drawing http://www.artyfactory.com/perspective_drawing/perspective_11.html
12. Shapes are found everywhere we go. You cannot escape them no matter what subject you get into. In biology you study animals and bugs that are all made of geometric shapes. Knowing geometric shapes would defiantly make their work easier.
Bibliography
http://www.funtrivia.com/askft/Question682.html
www.lung.ca/children/grades4_6/respiratory/insects.html
http://www.ehow.com/info_8111246_two-insect-mouth-parts-used.html
www.mathsisfun.com/geometry/polygons.html
http://www.aaamath.com/B/geo318x4.html
http://www.mathsisfun.com/geometry/interior-angles-polygons.html
...

...my family (guys) try and tell me to do something they say things like “no guy is gonna want a girl who can’t cook or want a girl who bites her nails or want a girl with such an attitude” as if my only purpose for living was to please some guy
I also need feminism because when my little cries, gets upset, or happy or expresses any emotion he’s told by family members (also guys) to stop acting like a girl. As if being a means keeping things bottled up and being tough all the time or that being a girl or even a little feminine is bad
Girls and boys are expected to act a certain way always to please someone else or to be someone else. To act like anything but them. And that is why I need gender equality. And why boys and girls need feminismmy family (guys) try and tell me to do something they say things like “no guy is gonna want a girl who can’t cook or want a girl who bites her nails or want a girl with such an attitude” as if my only purpose for living was to please some guy
I also need feminism because when my little cries, gets upset, or happy or expresses any emotion he’s told by family members (also guys) to stop acting like a girl. As if being a means keeping things bottled up and being tough all the time or that being a girl or even a little feminine is bad
Girls and boys are expected to act a certain way always to please someone else or to be someone else. To act like anything but them. And that is why I need gender equality. And why boys and girls need...

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