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...

...CHAPTER 7 ARITHMETIC AND GEOMETRIC PROGRESSIONS
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 sum,
(iii) a sum and another sum,
(iv) the definition of arithmetic progression.
(v)
Example 7-4
The third term of an arithmetic progression is 14 and the sixth term is 29....

...Questions from Questionbank
Topic 1. Sequences and Series, Exponentials and The Binomial Theorem
1. Find the sum of the arithmetic series
17 + 27 + 37 +...+ 417.
2. Find the coefficient of x5 in the expansion of (3x – 2)8.
3. An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series.
4. Find the coefficient of a3b4 in the expansion of (5a + b)7.
5. Solve the equation 43x–1 = 1.5625 × 10–2.
6. In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term.
7. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for
(a) log2 5;
(b) loga 20.
8. Find the sum of the infinite geometric series
9. Find the coefficient of a5b7 in the expansion of (a + b)12.
10. The Acme insurance company sells two savings plans, Plan A and Plan B.
For Plan A, an investor starts with an initial deposit of $1000 and increases this by $80 each month, so that in the second month, the deposit is $1080, the next month it is $1160 and so on.
For Plan B, the investor again starts with $1000 and each month deposits 6% more than the previous month.
(a) Write down the amount of money invested under Plan B in the second and third months.
Give your answers to parts (b) and (c) correct to the nearest dollar.
(b) Find the amount of the 12th deposit for each Plan.
(c) Find the total amount of money invested during the first 12 months
(i) under Plan A;...

...… 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 geometric series 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 asymptotes of f.
(2)
5.2
Calculate the intercepts of the graph of f with the axes.
(3)
5.3...

...Assessment 09.08 Geometric Series 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 that you can see...

...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 geometric series. 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 bottom row, how many total cans are on the bottom 14 rows? (1 point)
To solve you...

...Lang, Period 1
February 24, 2015
ProgressProgress takes time. Progress helps people change for the better, as a whole individual,
nation, or even world. Progress is the change that we want to see ourselves’s and in the world.
We do not know what change will bring and when it will happen, but we do know that progress
is our favorite challenge. Now as we live in a world with infinite history, we do not reach for a
finite end; we constantly pursue improvements to which there is no limit. In the course of doing
so, progress has become our favorite word. I believe Churchill's description of progress is correct
because progress to see the smallest of things.
I view progress as an ongoing task, it never ends and we don't ever stop pursuing it. No
matter what, progress is inevitable. For example, nobody would've ever guessed that slavery, and
it's time, could be ceased. The thought if African-Americans getting the right to vote was a
lunatic thought. However, with each day, the African-Americans got closer to freedom from
white oppression. This was “fruitful” for both sides of the argument, the proslavery South and
the anti-slavery North because we became a untied nation. However, racism still exists today and
the climb is not over for African-Americans. They still have some black stereotypes placed on
them, however when people try to stop...

...
This work MAT 126 Week 1 Assignment - Geometric and Arithmetic Sequence shows "Survey of Mathematical Methods" and contains solutions on the following problems:
First Problem: question 35 page 230
Second Problem: question 37 page 230
Mathematics - General Mathematics
Week One Written Assignment
Following completion of your readings, complete exercises 35 and 37 in the “Real World Applications” section on page 280 of Mathematics in Our World .
For each exercise, specify whether it involves an arithmetic sequence or a geometric sequence and use the proper formulas where applicable . Format your math work as shown in the Week One Assignment Guide and be concise in your reasoning. Plan the logic necessary to complete the exercise before you begin writing. For an example of the math required for this assignment, please review the Week One Assignment Guide .
The assignment must include ( a ) all math work required to answer the problems as well as ( b ) introduction and conclusion paragraphs.
Your introduction should include three to five sentences of general information about the topic at hand.
The body must contain a restatement of the problems and all math work, including the steps and formulas used to solve the problems.
Your conclusion must comprise a summary of the problems and the reason you selected a particular method to solve them. It would also be appropriate to include a...

...The Pilgrim’s Progress
By John Bunyan
20 April 2010
Format: MLA Style
The Pilgrims Progress, composed in 1678 by John Bunyan, is said to have originally graced John in a dream. As a Preacher and English writer, Bunyan comprised this during the time in which he was imprisoned for preaching the word of God. This makes good sense because of the timing of it all. If there were ever to be a good time for a person to consider their life as it was and eventual death that would one day come, it would be the time in which they were imprisoned if they were ever to find themselves in such a situation. Bunyan seemingly wrote this allegorical story to track the main character’s journey that would eventually lead him to find his salvation. As the author uses an allegorical style, he apologizes for it in the preface of the text but it actually saves the reader. Allegory uses symbolism as a disguised representation for meanings. Without allegory, the characters would have names that could easily take on the persona of any one person. The characters that Bunyan utilizes in this piece truly appear to be universal. The personalities of the characters that were conveyed could have been found just as easily in 1678 as they could in the present day. Carl Rollyson states that John Bunyan was a Puritan who wrote about every earnest Christian’s continuous search for salvation (394). The primary purpose was not only to spread the word, but to continue to...