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Brilio.net - Basically, a geometric series is a series in which each subsequent term is obtained by multiplying the previous term by a fixed number called the ratio. Understanding geometric series is important because this concept is the basis for understanding other more complex patterns in mathematics and science.
Why is it important to understand geometric series? Because this concept is widely used in various fields, including economics, physics, and computer science. For example, in economics, geometric series are used to calculate compound interest.
In physics, this series helps in calculations involving radioactive decay. While in computer science, this concept is used in search algorithms and certain data structures. Therefore, having a solid understanding of geometric series will make it easier to learn and apply further concepts.
To help you understand and master this concept, brilio.net will present examples of geometric series questions complete with formulas and basic understandings adapted from various sources, Thursday (5/9). Each example question is arranged to make it easier for you to understand the patterns in geometric series and how the formulas are applied.
Basic understanding of geometric series
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A geometric series is a series of numbers in which there is a constant ratio between any two consecutive terms. For example, in the simple geometric series 2, 4, 8, 16, and so on, the ratio between any two consecutive terms is 2. This means that each term is obtained by multiplying the previous term by 2.
The general formula for the nth term of a geometric series is:
[ a_n = a_1 times r^{(n-1)} ]
Where:
- ( a_n ) is the nth term,
- ( a_1 ) is the first term,
- ( r ) is the series ratio,
- ( n ) is the term number.
To calculate the sum of the first n terms of a geometric series, use the formula:
[ S_n = frac{a_1 times (r^n - 1)}{r - 1} ]
If ( r = 1 ), then the formula is ( S_n = n times a_1 ).
After understanding the definition and formula of geometric series, you can find out more about geometric series by looking at several examples that brilio.net has collected from various sources, Thursday (5/9).
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1. Question 1
Given a geometric series: 3, 6, 12, ... Determine the 5th term of the series!
Answer:
Use the formula ( a_n = a_1 times r^{(n-1)} ). Given ( a_1 = 3 ), ( r = 2 ), ( n = 5 ).
Then, ( a_5 = 3 times 2^{4} = 3 times 16 = 48 ).
2. Question 2
Determine the sum of the first 4 terms of the series 1, 2, 4, 8, ...
Answer:
Given (a_1 = 1), (r = 2), (n = 4).
Then, ( S_4 = frac{1 times (2^4 - 1)}{2 - 1} = frac{1 times 15}{1} = 15 ).
3. Question 3
If the third term of a geometric series is 16 and the fifth term is 64, determine the ratio ( r ) and the first term ( a_1 )!
Answer:
It is known that (a_3 = 16), (a_5 = 64).
Use the formula ( a_n = a_1 times r^{(n-1)} ).
( 16 = a_1 times r^2 )
( 64 = a_1 times r^4 )
So, ( r = sqrt{4} = 2 ), and ( a_1 = 16 div 4 = 4 ).
4. Question 4
What is the sum of the first 6 terms of a geometric series that has ( a_1 = 5 ) and ( r = 3 )?
Answer:
Use the formula ( S_n = frac{a_1 times (r^n - 1)}{r - 1} ).
( S_6 = frac{5 times (3^6 - 1)}{2} = frac{5 times 728}{2} = 1820 ).
5. Question 5
Determine the 10th term of the series 1, -2, 4, -8, ...
Answer:
Given (a_1 = 1), (r = -2), (n = 10).
Then, ( a_{10} = 1 times (-2)^9 = -512 ).
6. Question 6
If the first term of a geometric sequence is 7 and the ratio is 3, calculate the 7th term!
Answer:
Given (a_1 = 7), (r = 3), (n = 7).
Then, ( a_7 = 7 times 3^6 = 5103 ).
7. Question 7
Calculate the sum of the first 5 terms of the series 2, 6, 18, 54, ...
Answer:
Given (a_1 = 2), (r = 3), (n = 5).
Then, ( S_5 = frac{2 times (3^5 - 1)}{2} = 484 ).
8. Question 8
If the first term of the series is 4 and the ratio is 0.5, calculate the 4th term!
Answer:
It is known that (a_1 = 4), (r = 0.5), (n = 4).
Then, ( a_4 = 4 times (0.5)^3 = 0.5 ).
9. Question 9
What is the sum of the first 4 terms of the series 10, 5, 2.5, ...?
Answer:
It is known that (a_1 = 10), (r = 0.5), (n = 4).
Then, ( S_4 = frac{10 times (0.5^4 - 1)}{0.5 - 1} = 18.75 ).
10. Question 10
Determine the 3rd term of a geometric series whose first term is 8 and the ratio is 1/3!
Answer:
Given ( a_1 = 8 ), ( r = frac{1}{3} ), ( n = 3 ).
Then, ( a_3 = 8 times (frac{1}{3})^2 = frac{8}{9} ).
(brl/red)
