**Geometry Series –** Discussion of material about arithmetic sequences and series, you will definitely learn along with material on geometric sequences. Even though they look the same, the two materials have their own characteristics and formulas.

The difference between arithmetic sequences and series and geometric sequences and series is the pattern. If in arithmetic we use addition patterns, then in geometry we use multiplication patterns. Well, like material in other branches of science, the higher the level of discussion, the more difficult it will be. But don’t worry, because *Sinaumed’s* will still understand it all if he understands the concept of the formula.

So, what is a geometric series? What is the concept of the formula for this geometric series? What about examples of problems regarding geometric series and their discussion? So, so that *Sinaumed’s* isn’t confused about these things, let’s immediately see the following review!

## What is a Geometry Series?

According to Ruangguru , a geometric series is one that looks like a geometric sequence, but is written in addition form. The ratio in the geometric series is symbolized by r. A simple example of a geometric series is: 1 + 4 + 16 + 64 + 256,….

Yep, the thing that distinguishes a geometric sequence from a geometric series is the way of writing the arrangement. If in a geometric sequence, the numbers are separated using a comma (,), then in a geometric series using an addition sign (+). **“That is why, the definition of a geometric series is the sum of each term of a geometric sequence.”**

In order to understand better, pay attention to writing the following standard arrangement patterns for geometric sequences and geometric series!

Geometry sequence: a, ar, ar2 , ar3 , …, arn – 1 |

Geometry series: a + ar + ar2 + ar3 + … + arn – 1 |

Well, based on various sources it can be concluded about matters regarding geometric series, namely.

- A geometric series is the sum of the terms in the geometric sequence.
- The sum in question is the sum for several finite terms (starting from the first n terms).
- The symbol used is Sn, meaning the number of first n terms.

Another example of a geometric series is:

S _{1} = U _{1} (sum of 1 first term)

S _{2} = U _{1} + U _{2} (sum of the first 2 terms)

S _{3} = U _{1} + U _{2} + U _{3} (sum of the first 3 terms)

S _{4} = U _{1} + U _{2} + U _{3} + U 4 (sum of first 4 terms)

etc.

### Understanding What An Infinite Geometry Series Is

The discussion of geometric series will also be related to infinite geometric series, of course the sum will reach the infinity term. The number of the series still follows the geometric sequence. Since this geometric series is infinite, it will use the symbol ∞ alias infinity (infinite).

## Geometry Series Formulas

The formula for this geometric series is of course different from the formula for an arithmetic series, even with the formula for an infinite geometric series. This is because, even though these three things have the same name as “series”, their definitions and formulas will still be different. The following is a formula for calculating a geometric series!

**Ascending series (r > 1)**

**Descending series (r < 1)**

**Information:**

Sn = Sum of the nth term of the geometric series

a = First term

r = Ratio

### Proof of Geometry Series Formulas

The following is a proof of the geometric series formula, especially in the descending series for r < 1.

⇔ Sn = U _{1} + U _{2} + U _{3} + U 4 + … + Un

⇔ = a + ar + ar ^{2} + ar ^{3} + …+ ar ^{n-1} ……………………… (1)

Now, from equation (1), all terms will be multiplied by r, so it becomes:

⇔ rS _{n} = r (U _{1} + U _{2} + U _{3} + U 4 + … + Un )

⇔ = r ( a + ar + ar ^{2} + ar ^{3} + …+ ar ^{n-1} )

⇔ = a + ar + ar ^{2} + ar ^{3} + …+ ar ^{n} ………………… (2)

Then, from equations (1) and (2), the following calculation will be obtained:

S _{n} = a + ar + ar ^{2} + ar ^{3} + …+ ar ^{n-1}

rS _{n} = ar + ar ^{2} + ar ^{3} + ar ^{4} +….. + ar ^{n}

————————————————————— –

S _{n} – rS _{n} = a + (-ar ^{n} )

(1-r) S _{n} = a – ar ^{n}

## Examples of Geometry Series Problems and their Discussion

### Example Question 1

- Find the sum of the first 9 terms of the geometric series 3 + 6 + 12 + 24 + 48 + …

**Completion:**

Given: a = 3

Wanted: S _{9}

Answer:

### Example Problem 2

2. A rope is divided into 6 parts with length measurements to form a geometric series; if the shortest part is 3 cm and the longest is 96 cm, determine the length of the rope.

Completion:

Given: Un = 96; a = 3; n = 6

Asked: S _{7}

Answer:

Un = ar ^{n-1}

⇔ 96 = 3 . r _{5}

⇔ r _{5} = 32

⇔r = 2

Because r > 1, the applicable calculation formula is

So, the length of the rope is 189 cm.

## 15+ Geometry Series Problems

- It is known that the sequence √3 , 3, 3√3 , … The 9th term is …

A. 81√3 | B. 81 | C. 243 | D. 613√3 | E.729 |

- In a geometric sequence, it is known that the 3rd term is 3 and the 6th term is 81. Then the 8th term is…

A. 729 | B. 612 | C. 542 | D. 712 | E. 681 |

- You know the sequence 2, 2 2 , 4, 4 2 , … What term is 64√2?

A.11 | B.12 | C. 13 | D. 14 | E. 15 |

- The sum of the first 5 terms of the series 3 + 6 + 12 + … is …

A. 62 | B. 84 | C. 93 | D. 108 | E. 152 |

- The sum of the first n terms of a geometric series is expressed by Sn = 2n+2 – 3. The formula for the nth term is…

A. . 2n–1 | B. 2n+1 | C. 2n+3 | D. . 2n–3 | E. 2n |

- It is known that a geometric series with the first term is 6 and the fourth term is 48. The sum of the first six terms of the series is …

A.368 | B.389 | C. 378 | D. 379 | E.384 |

- Given four numbers, the first three numbers are arithmetic sequences and the last three numbers are geometric sequences. The sum of the second and fourth numbers is 8. The sum of the first and third numbers is 18. The sum of the four numbers is…

A.28 | B. 31 | C. 44 | D. 52 | E. 81 |

- A rope is cut into 8 pieces. The length of each of these pieces follows the geometric sequence. The length of the shortest piece of rope is 4 cm and the length of the longest piece of rope is 512 cm. The length of the original rope is … cm

A.512 | B. 1020 | C. 1024 | D. 2032 | E.2048 |

- The following series is known: 3 + 9 + 27 + 81 + …

- Determine the 8th term in the series!
- Determine the sum of the first 8 terms in the series!

- Bacteria reproduce by dividing every 30 minutes. If the number of bacteria is 200, count how many bacteria will grow after 12 hours and after 24 hours!
- Calculate the sum of the geometric series: 3 + 6 + 12 + …. +384

## Understanding What an Arithmetic Sequence Is

### What is the Arithmetic Sequence Formula?

Please note, *Sinaumed’s,* that the formulas for arithmetic sequences and arithmetic series are different, even though both are sub-chapters of the same material. Well, here is the formula for calculating an arithmetic sequence.

Un = a + (n – 1)b |

Information:

a = U1 = first term in the arithmetic sequence

b = difference in arithmetic sequence = Un – Un-1, provided that n is the number of terms

n = number of terms

Un = the number of the nth term

### The formula for finding the difference in an arithmetic sequence

b = Un – Un-1 |

**Information:**

b = difference in the arithmetic sequence

Un = the nth term

Un-1 – the n-1 term

## Examples of Problem Arithmetic Sequences and their Discussion

### Example Question 1

Find the 100th term of the arithmetic sequence 2, 5, 8, 11, …

**Discussion:**

a = 2

b = u2 – u1 = 5 – 2 = 3

n = 100 un = a + (n – 1)b

un = 2 + (100 – 1)3 = 2 + (99 x 3) = 299

### Example Problem 2

The arithmetic sequence 1, 3, 5, 7, … is known. un = 225. Determine the number of terms (n).

**Completion:**

a = 1, b = 2, un = 225

un = a (n – 1)b

225 = 1 + (n – 1)2 = 1 + 2n – 2

226 = 2n

n = 113

### Example Problem 3

Si Dadap successfully passed the PT (Higher Education) entrance examination. As a student, starting January 1, 2008 he received an allowance of Rp. 500,000.00 for one quarter. This allowance is given at the beginning of each quarter. For each subsequent quarter the pocket money he receives is increased by Rp. 25,000. How much pocket money will Dadap receive in early 2011?

**Completion:**

1st Quarter: u1 = a = Rp. 500,000.00

2nd quarter: u2 = a + b = Rp. 525,000.00, etc

So b = 25,000.

At the beginning of 2011, the college had been used for 3 years or 12 quarters, meaning: u12 = a + (12 – 1)b = 500,000 + (11 x 25,000) = 775,000

So the amount of money that Dadap will receive at the beginning of 2011 is Rp. 775,000.00.

### Example Problem 4

It is known that the 1st term of the arithmetic sequence is 6 and the fifth term is 18, determine the difference.

**Completion: **

It is known that a = 6, and U5 = 18

Un = a + (n – 1) b

U5 = 6 + (5 – 1) b

18= 6 + 4b

4b = 12

b = 3

So the difference is 3.

### Example Problem 5

Find the 21st term of the arithmetic sequence: 17, 15, 13, 11,…

**Completion: **

If a = 17, b = -2, and n = 21,

then U21 = 17 + (21-1)(-2) = **-23 **

So, the 21st term of the arithmetic sequence is -23

### Example Problem 6

The 40th term of sequence 7, 5, 3, 1, … is …

**Completion:**

Given: a = 7

b = –2

Asked 𝑈40 ?

Answer:

𝑈𝑛 = 𝑎 + (𝑛 − 1) 𝑏

𝑈40 = 7 + (40 − 1) (−2)

= 7 + 39 x (-2)

= 7 + (-78) = – 71

So, the 40th term of the arithmetic sequence is –71.

### Example Problem 7

The formula for the nth term of the sequence 5, –2, –9, –16, … is …

**Discussion: **

Given: a = 5 b = –7

Wanted: the formula for the nth term of the arithmetic sequence = ?

**Answer:**

𝑈𝑛 = 𝑎 + (𝑛 − 1) 𝑏

= 5 + (𝑛 − 1)(−7)

= 5 − 7 𝑛 + 7

= 12 − 7 𝑛

So, the formula for the nth term of the arithmetic sequence is 𝑈𝑛 = 12 − 7𝑛

### Example Problem 8

In a theater, seats are arranged with the front row consisting of 12 seats, the second row containing 14 seats, the third row containing 16 seats, and so on. The number of seats in the 20th row is…

**Discussion: **

Is known:

a = 12

b = 2

Asked 𝑈20 ?

**Answer:**

𝑈𝑛 = 𝑎 + (𝑛 − 1)𝑏

𝑈20 = 12 + (20 − 1)(2)

= 12 + 19 . (2)

= 12 + (38) = 50

So, the number of seats in the 20th row is 50 seats

### Example Problem 9

The 10th number of the sequence: 3, 5, 7, 9, ….is…

**Completion:**

a = 3, b = 2,

U10 = (a + 9b)

U10 = 3 + 18 = 21

### Example Problem 10

A sequence of 2, 5, 10, 17, …. meet the pattern Un = an2 + bn + c. 9th tribe of

that line is…

**Completion**

Is known :

Rows 2, 5, 10, 17, …

𝑈𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐

Asked: 𝑈9 = ⋯ ?

**Answer:**

𝑈𝑛 = (1)𝑛 ^{2} + (0)𝑛 + 1

𝑈𝑛 = 𝑛2 + 1

𝑈9 = 92 + 1

𝑈9 = 82

**Source:**

Dhoruri, Atmini. Sequences and Series of Numbers.

Istiqomah. (2020). *General Mathematics High School Learning Module: Class XI Sequences and Series* . Ministry of Education and Culture. SMA Negeri 5 Mataram.

Karso, H. *Lines and Derets (High School Mathematics Learning)* . FMIPA UPI.

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