Calculating Sum Of 10 Terms In Arithmetic And Geometric Sequences
This article delves into the fascinating world of sequences, focusing on calculating the sum of the first 10 terms in various mathematical progressions. We'll explore arithmetic progressions (APs) and geometric progressions (GPs), applying specific formulas to arrive at our solutions. Understanding these concepts is crucial for various fields, from finance to physics, making this exploration both practical and intellectually stimulating.
i) 1/2, 1/3, 2/9
In this first sequence, identifying the pattern is crucial for calculating the sum of 10 terms. We observe the sequence 1/2, 1/3, and 2/9. A closer look reveals that this sequence is a geometric progression (GP). In a GP, each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. To confirm this, we need to determine the common ratio (r) and the first term (a).
The first term (a) of the sequence is clearly 1/2. To find the common ratio (r), we divide the second term by the first term (or the third term by the second term). Thus, r = (1/3) / (1/2) = 2/3. We can verify this by checking if (2/9) / (1/3) also equals 2/3, which it does. Therefore, we have a GP with a = 1/2 and r = 2/3.
Now, to calculate the sum of the first 10 terms (S10) of this GP, we use the formula for the sum of n terms of a GP: Sn = a(1 - r^n) / (1 - r), where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. In our case, a = 1/2, r = 2/3, and n = 10. Plugging these values into the formula, we get:
S10 = (1/2) * (1 - (2/3)^10) / (1 - 2/3)
Simplifying this expression, we first calculate (2/3)^10, which is approximately 0.01734. Then, 1 - (2/3)^10 ≈ 1 - 0.01734 = 0.98266. Next, we calculate the denominator, 1 - 2/3 = 1/3. Finally, S10 = (1/2) * 0.98266 / (1/3) = (1/2) * 0.98266 * 3 = 1.47399. Therefore, the sum of the first 10 terms of the sequence 1/2, 1/3, 2/9 is approximately 1.474.
Understanding the formula for the sum of a GP is paramount in solving such problems. The formula elegantly captures the essence of geometric progression, allowing us to efficiently calculate the sum of a large number of terms without having to individually add them. Moreover, this concept extends beyond pure mathematics, finding applications in financial calculations such as compound interest and annuity calculations.
ii) 12, 1, 1/12
Moving on to the second sequence, 12, 1, 1/12, we need to determine if it's an arithmetic progression (AP) or a geometric progression (GP). An AP has a constant difference between consecutive terms, while a GP has a constant ratio. Examining the sequence, we see that the difference between the first two terms (1 - 12 = -11) is not the same as the difference between the second and third terms (1/12 - 1 = -11/12). Therefore, it's not an AP.
However, if we look at the ratio between consecutive terms, we find that 1/12 = 1/12 and (1/12)/1 = 1/12. This constant ratio indicates that the sequence is a geometric progression (GP). The first term (a) is 12, and the common ratio (r) is 1/12.
To find the sum of the first 10 terms (S10), we again use the formula for the sum of n terms of a GP: Sn = a(1 - r^n) / (1 - r). Substituting a = 12, r = 1/12, and n = 10, we get:
S10 = 12 * (1 - (1/12)^10) / (1 - 1/12)
Let's break down the calculation. First, we calculate (1/12)^10, which is an extremely small number, approximately 1.656 x 10^-11. For practical purposes, we can consider it to be almost zero. Then, 1 - (1/12)^10 ≈ 1 - 0 = 1. The denominator becomes 1 - 1/12 = 11/12. Thus, S10 = 12 * 1 / (11/12) = 12 * (12/11) = 144/11. Converting this to a decimal, we get approximately 13.09.
Therefore, the sum of the first 10 terms of the sequence 12, 1, 1/12 is approximately 13.09. The fact that (1/12)^10 is so small highlights a key characteristic of GPs with a common ratio less than 1 in absolute value: as n increases, r^n approaches zero, and the sum of the series converges to a finite value. This concept is fundamental in areas like infinite series and calculus.
iii) 1/2, 1, 2, 4
Considering the third sequence, 1/2, 1, 2, 4, we once again determine whether it is an AP or a GP. The difference between the first two terms is 1 - 1/2 = 1/2, and the difference between the second and third terms is 2 - 1 = 1. Since these differences are not equal, it is not an AP.
Now, let's examine the ratios. The ratio between the second and first terms is 1 / (1/2) = 2, and the ratio between the third and second terms is 2 / 1 = 2. Similarly, the ratio between the fourth and third terms is 4 / 2 = 2. This constant ratio confirms that the sequence is a geometric progression (GP). Here, the first term (a) is 1/2, and the common ratio (r) is 2.
Using the formula for the sum of n terms of a GP, Sn = a(1 - r^n) / (1 - r), with a = 1/2, r = 2, and n = 10, we can calculate the sum of the first 10 terms (S10):
S10 = (1/2) * (1 - 2^10) / (1 - 2)
First, we calculate 2^10, which is 1024. Then, 1 - 2^10 = 1 - 1024 = -1023. The denominator is 1 - 2 = -1. Therefore, S10 = (1/2) * (-1023) / (-1) = (1/2) * 1023 = 511.5.
Thus, the sum of the first 10 terms of the sequence 1/2, 1, 2, 4 is 511.5. This example illustrates a GP where the common ratio is greater than 1. In such cases, the terms of the sequence grow exponentially, and the sum of the series also increases rapidly. This principle is applied in various exponential growth models across disciplines like biology and economics.
iv) 12, 24, 48
Finally, let's analyze the sequence 12, 24, 48. The difference between the first two terms is 24 - 12 = 12, and the difference between the second and third terms is 48 - 24 = 24. As the differences are not the same, this is not an AP.
However, if we examine the ratios, we find that 24 / 12 = 2 and 48 / 24 = 2. This consistent ratio indicates that the sequence is a geometric progression (GP). The first term (a) is 12, and the common ratio (r) is 2.
To determine the sum of the first 10 terms (S10), we apply the GP sum formula: Sn = a(1 - r^n) / (1 - r). Substituting a = 12, r = 2, and n = 10, we get:
S10 = 12 * (1 - 2^10) / (1 - 2)
We already know that 2^10 is 1024. Therefore, 1 - 2^10 = 1 - 1024 = -1023. The denominator is 1 - 2 = -1. Thus, S10 = 12 * (-1023) / (-1) = 12 * 1023 = 12276.
Therefore, the sum of the first 10 terms of the sequence 12, 24, 48 is 12276. This example further demonstrates the rapid growth characteristic of GPs with a common ratio greater than 1. The sum increases dramatically as more terms are included, highlighting the power of exponential growth and its relevance in real-world scenarios.
In conclusion, by carefully analyzing each sequence and applying the appropriate formulas for arithmetic and geometric progressions, we have successfully calculated the sum of the first 10 terms for each given series. This exercise underscores the importance of recognizing patterns in sequences and applying the correct mathematical tools to solve related problems. Understanding these concepts not only strengthens mathematical skills but also provides a foundation for tackling complex problems in various fields.
