Geometric Sequence Summation Problems And Solutions

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When delving into the realm of geometric sequences, one of the fundamental concepts is calculating the sum of its terms. A geometric sequence is characterized by a constant ratio between consecutive terms, often denoted as 'r,' and a first term, 'a₁.' The sum of the first 'n' terms of a geometric sequence can be determined using a specific formula, which we will explore in detail. Let's consider a geometric sequence where the first term (a₁) is 4, the common ratio (r) is 2, and we aim to find the sum of the first 20 terms (n = 20). Understanding the intricacies of this calculation provides a solid foundation for tackling more complex problems involving geometric sequences and series.

The formula for the sum (Sn) of the first n terms of a geometric sequence is given by:

Sn = a₁(1 - rⁿ) / (1 - r)

where:

  • a₁ is the first term,
  • r is the common ratio, and
  • n is the number of terms.

In this specific scenario, we have a₁ = 4, r = 2, and n = 20. Plugging these values into the formula, we get:

S₂₀ = 4(1 - 2²⁰) / (1 - 2)

Now, let's break down the calculation step by step:

  1. Calculate 2²⁰: 2²⁰ = 1,048,576
  2. Substitute this value back into the formula: S₂₀ = 4(1 - 1,048,576) / (1 - 2)
  3. Simplify the expression inside the parentheses: S₂₀ = 4(-1,048,575) / (-1)
  4. Multiply 4 by -1,048,575: S₂₀ = -4,194,300 / (-1)
  5. Finally, divide -4,194,300 by -1: S₂₀ = 4,194,300

Therefore, the sum of the first 20 terms of the geometric sequence with a first term of 4 and a common ratio of 2 is 4,194,300. This substantial sum highlights the exponential growth inherent in geometric sequences with a common ratio greater than 1. Each term in the sequence is twice the previous term, leading to a rapid increase in the magnitude of the terms and, consequently, the sum. Understanding this principle is crucial for predicting the behavior of geometric sequences and series in various mathematical and real-world applications.

Let's further solidify our understanding of geometric series summation by considering another example. This time, we have a geometric sequence with a first term (a₁) of -5, a common ratio (r) of 3, and we wish to determine the sum of the first 9 terms (n = 9). This example provides an opportunity to apply the same formula we used previously, but with different values, allowing us to appreciate the versatility of the formula and its applicability across various geometric sequences. The negative first term and a common ratio greater than 1 introduce interesting dynamics into the sequence, influencing the pattern of the terms and the overall sum. Understanding how these factors interact is key to mastering geometric sequence calculations.

Again, we utilize the formula for the sum (Sn) of the first n terms of a geometric sequence:

Sn = a₁(1 - rⁿ) / (1 - r)

Plugging in the given values, a₁ = -5, r = 3, and n = 9, we have:

S₉ = -5(1 - 3⁹) / (1 - 3)

Now, let's break down the calculation:

  1. Calculate 3⁹: 3⁹ = 19,683
  2. Substitute this value into the formula: S₉ = -5(1 - 19,683) / (1 - 3)
  3. Simplify the expression inside the parentheses: S₉ = -5(-19,682) / (-2)
  4. Multiply -5 by -19,682: S₉ = 98,410 / (-2)
  5. Finally, divide 98,410 by -2: S₉ = -49,205

Therefore, the sum of the first 9 terms of the geometric sequence with a first term of -5 and a common ratio of 3 is -49,205. The negative sum arises from the negative first term and the fact that the common ratio is greater than 1, leading to terms that alternate in sign but grow rapidly in magnitude. The negative sign in the result indicates that the sum of the negative terms outweighs the sum of the positive terms in the sequence. This example underscores the importance of carefully considering the signs of the terms and the common ratio when calculating the sum of a geometric sequence.

Now, let's explore a geometric sequence where the common ratio is a fraction. This scenario introduces a different dynamic compared to sequences with integer common ratios. Consider a geometric sequence with a first term (a₁) of -216, a common ratio (r) of 1/2, and we want to find the sum of the first 5 terms (n = 5). A fractional common ratio, specifically one between 0 and 1, signifies that the terms of the sequence will decrease in magnitude as we progress through the sequence. This diminishing effect has a significant impact on the sum of the terms, potentially leading to a convergence towards a finite value as the number of terms increases. Understanding this behavior is crucial when dealing with infinite geometric series.

Using the formula for the sum (Sn) of the first n terms of a geometric sequence:

Sn = a₁(1 - rⁿ) / (1 - r)

Plugging in the values a₁ = -216, r = 1/2, and n = 5, we get:

S₅ = -216(1 - (1/2)⁵) / (1 - 1/2)

Let's break down the calculation step by step:

  1. Calculate (1/2)⁵: (1/2)⁵ = 1/32
  2. Substitute this value into the formula: S₅ = -216(1 - 1/32) / (1 - 1/2)
  3. Simplify the expressions inside the parentheses: S₅ = -216(31/32) / (1/2)
  4. Multiply -216 by 31/32: S₅ = -6696/32 / (1/2)
  5. Divide -6696/32 by 1/2 (which is the same as multiplying by 2): S₅ = -6696/32 * 2 = -6696/16
  6. Simplify the fraction: S₅ = -418.5

Thus, the sum of the first 5 terms of the geometric sequence with a first term of -216 and a common ratio of 1/2 is -418.5. The negative sum reflects the negative first term, while the fractional common ratio causes the terms to decrease in magnitude, leading to a sum that is less extreme than in the previous examples with integer common ratios greater than 1. This example demonstrates the impact of a fractional common ratio on the sum of a geometric sequence, highlighting the concept of convergence as the terms approach zero.

Geometric sequences are defined by a constant ratio between consecutive terms. To find the sum of the first 'n' terms, we need to identify the first term (a₁), the common ratio (r), and the number of terms (n). In this case, we're given the first few terms of the sequence: 6, 24, -96, 384, ... and we want to find the sum of the first 7 terms (n = 7). The alternating signs in this sequence suggest a negative common ratio, which will influence the overall sum. Understanding how to extract the necessary parameters from a given sequence is a fundamental skill in working with geometric sequences and series.

First, let's find the common ratio (r) by dividing any term by its preceding term. For instance, dividing the second term (24) by the first term (6): r = 24 / 6 = 4. However, dividing the third term (-96) by the second term (24): r = -96 / 24 = -4. The fourth term (384) by the third term (-96): r = 384 / -96 = -4. This confirms that the common ratio r = -4. The first term (a₁) is clearly 6.

Now we have a₁ = 6, r = -4, and n = 7. We can use the formula for the sum of the first n terms of a geometric sequence:

Sn = a₁(1 - rⁿ) / (1 - r)

Substitute the values into the formula:

S₇ = 6(1 - (-4)⁷) / (1 - (-4))

Let's calculate step by step:

  1. Calculate (-4)⁷: (-4)⁷ = -16384
  2. Substitute this value into the formula: S₇ = 6(1 - (-16384)) / (1 - (-4))
  3. Simplify the expressions inside the parentheses: S₇ = 6(1 + 16384) / (1 + 4)
  4. Continue simplifying: S₇ = 6(16385) / 5
  5. Multiply 6 by 16385: S₇ = 98310 / 5
  6. Finally, divide 98310 by 5: S₇ = 19662

Therefore, the sum of the first 7 terms of the geometric sequence 6, 24, -96, 384, ... is 19662. The positive sum, despite the alternating signs of the terms, indicates that the positive terms contribute more significantly to the overall sum. This example demonstrates the impact of a negative common ratio on the pattern of terms and the resulting sum, highlighting the interplay between the magnitude and signs of the terms.

In this final example, we'll work with a geometric sequence that features decimal terms. This scenario is particularly relevant in real-world applications where quantities are often measured with decimal precision. We're given the sequence 1.6, 0.08, 0.004, ... and we want to find the sum of the first 5 terms (n = 5). The decreasing magnitude of the terms suggests a common ratio between 0 and 1, which will lead to a sum that converges as we add more terms. This example provides an opportunity to practice working with decimals in the context of geometric sequences and series, further solidifying our understanding of the core concepts.

To determine the common ratio (r), we divide any term by its preceding term. Dividing the second term (0.08) by the first term (1.6): r = 0.08 / 1.6 = 0.05. Dividing the third term (0.004) by the second term (0.08): r = 0.004 / 0.08 = 0.05. This confirms that the common ratio r = 0.05. The first term (a₁) is 1.6.

Now we have a₁ = 1.6, r = 0.05, and n = 5. Using the formula for the sum of the first n terms of a geometric sequence:

Sn = a₁(1 - rⁿ) / (1 - r)

Substitute the values:

S₅ = 1.6(1 - (0.05)⁵) / (1 - 0.05)

Let's break down the calculation:

  1. Calculate (0.05)⁵: (0.05)⁵ = 0.0000003125
  2. Substitute this value: S₅ = 1.6(1 - 0.0000003125) / (1 - 0.05)
  3. Simplify the expressions inside the parentheses: S₅ = 1.6(0.9999996875) / (0.95)
  4. Multiply 1.6 by 0.9999996875: S₅ = 1.5999995 / 0.95
  5. Finally, divide 1.5999995 by 0.95: S₅ ≈ 1.68421

Therefore, the sum of the first 5 terms of the geometric sequence 1.6, 0.08, 0.004, ... is approximately 1.68421. The sum is slightly larger than the first term due to the small, positive contribution of the subsequent terms. The common ratio of 0.05, being much smaller than 1, causes the terms to decrease rapidly, resulting in a sum that converges quickly. This example illustrates the application of the geometric series summation formula in a scenario involving decimal terms and a common ratio that leads to convergence.

In conclusion, understanding and applying the formula for the sum of a geometric sequence is a fundamental skill in mathematics. By working through these diverse examples, we've gained a deeper appreciation for the interplay between the first term, the common ratio, and the number of terms in determining the sum. From sequences with integer common ratios greater than 1, which exhibit rapid growth, to sequences with fractional common ratios that converge, the formula provides a powerful tool for calculating the sum of a wide range of geometric sequences. The ability to analyze and manipulate geometric sequences is not only essential in mathematics but also finds applications in various fields, including finance, physics, and computer science. Mastering these concepts opens doors to understanding more complex mathematical models and real-world phenomena.