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...

...× 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...

...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...

...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...

...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...

...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....

...
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 ....

...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...