# Geometric Progression Series and Sums

Good Essays
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, ...

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

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

Finding the sum of a Geometric Series to Infinity
Question

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

## You May Also Find These Documents Helpful

• Good Essays

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…

• 378 Words
• 2 Pages
Good Essays
• Satisfactory Essays

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,…

• 2486 Words
• 10 Pages
Satisfactory Essays
• Satisfactory Essays

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,…

• 731 Words
• 6 Pages
Satisfactory Essays
• Good Essays

What is the closed-form expression for the below sum of Geometric Progression (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…

• 696 Words
• 3 Pages
Good Essays
• Satisfactory Essays

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……

• 379 Words
• 2 Pages
Satisfactory Essays
• Powerful Essays

Geometric Design of Highways -refer to the visible dimensions of streets and highways. -Its main purpose is to provide safe, efficient, and economical movement of traffic. Major areas of the design: Locational design Alignment design (includes design controls and criteria) Cross-section design Locational Design One of the most crucial and important parts of the design process of highways. The location procedure is an iterative process in which engineers, planners, economist, ecologist…

• 2882 Words
• 12 Pages
Powerful Essays
• Better Essays

Geometric Krater The Geometric Krater is a magnificent piece of Greek Art. In the eight century, vase painting became very popular. The vases show a great show a great variety of style and development over the centuries, beginning with the geometric and very linear style. They then continued through the oriental style which borrowed images from the eastern world, and into the classical era with mythology portrayed with as much classical accuracy as the ancient Greek potters and painters could…

• 866 Words
• 4 Pages
Better Essays
• Good Essays

Geometric mean From Wikipedia, the free encyclopedia Jump to: navigation, search The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the set, n, the numbers are multiplied and then the nth root of the resulting…

• 6930 Words
• 28 Pages
Good Essays
• Satisfactory Essays

TIME VALUE OF MONEY SUMS 1. A finance company advertises that it will pay a lumpsum of Rs 8000 at the end of 6 yrs to investors who deposit annually Rs 1000 for 6 yrs. What is the rate implicit in this offer? 2. You want to take a trip to the moon which costs Rs 10,00,000-the cost is expected to remain unchanged in nominal terms. You can save annually Rs 50000 to fulfil this desire. How long will you have to wait if your savings earn an interest of 12 percent p.a.? 3. Suppose a firm borrows…

• 364 Words
• 2 Pages
Satisfactory Essays
• Powerful Essays

VSW24:Painting Art Project Assessment 3 Drawing Automaton (Robot) http://www.youtube.com/watc h?v=PR_kFssrpms Geometric Abstraction by Machines - Final Work 1 Concept The purpose of this project is to investigate the use of home-made or repurposed machines to generate geometric patterns. Jackson Pollock redefined what it was to produce art. He removed prior boundaries to making art. His move away from conventionality was a liberating signal to future artists. It is my endeavour…

• 3023 Words
• 13 Pages
Powerful Essays