Geometric Sequence Combining Functions And Finding The 9th Term

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In the realm of mathematical sequences, geometric sequences hold a special place due to their consistent multiplicative pattern. These sequences are characterized by a constant ratio between consecutive terms, a property that allows us to predict any term in the sequence with ease. In this article, we will explore how to combine two given functions, f(n) = 11 and g(n) = (3/4)^(n-1), to construct a geometric sequence, denoted as a_n. Furthermore, we will delve into the process of calculating the 9th term of this newly formed sequence.

Understanding Geometric Sequences

Before we embark on the journey of combining functions, it is crucial to establish a firm understanding of geometric sequences. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the preceding term by a constant factor, known as the common ratio. This common ratio, often denoted by r, is the cornerstone of a geometric sequence, as it dictates the multiplicative relationship between successive terms.

For instance, consider the sequence 2, 4, 8, 16, 32... This is a geometric sequence with a common ratio of 2, as each term is twice the value of the preceding term. Similarly, the sequence 100, 50, 25, 12.5, 6.25... is a geometric sequence with a common ratio of 0.5, where each term is half the value of its predecessor.

The general formula for the nth term of a geometric sequence is given by:

a_n = a_1 * r^(n-1)

where:

  • a_n represents the nth term of the sequence
  • a_1 denotes the first term of the sequence
  • r signifies the common ratio
  • n represents the term number

This formula serves as a powerful tool for calculating any term in a geometric sequence, provided we know the first term and the common ratio.

Combining Functions to Form a Geometric Sequence

Now that we have a solid grasp of geometric sequences, let's turn our attention to the task at hand: combining the functions f(n) = 11 and g(n) = (3/4)^(n-1) to create a geometric sequence, a_n. The key to achieving this lies in recognizing that the function g(n) already embodies the essence of a geometric sequence, with its exponential form hinting at a constant multiplicative relationship.

The function f(n) = 11, on the other hand, represents a constant value, independent of the term number n. To seamlessly integrate this constant into a geometric sequence, we can consider it as the first term, a_1, of our desired sequence. This means that the geometric sequence a_n will initiate with the value 11.

To complete the construction of a_n, we need to incorporate the geometric nature of g(n). The function g(n) = (3/4)^(n-1) exhibits an exponential decay, indicating a common ratio of 3/4. This implies that each term in the sequence will be 3/4 times the value of the preceding term.

Therefore, we can combine f(n) and g(n) to define the geometric sequence a_n as follows:

a_n = f(n) * g(n) = 11 * (3/4)^(n-1)

This formula elegantly captures the essence of our geometric sequence, where the first term is 11 and the common ratio is 3/4. The term (3/4)^(n-1) ensures the geometric progression, while the constant factor of 11 scales the entire sequence.

Solving for the 9th Term

With the formula for a_n firmly established, we can now proceed to calculate the 9th term of the sequence. To achieve this, we simply substitute n = 9 into the formula:

a_9 = 11 * (3/4)^(9-1) = 11 * (3/4)^8

Evaluating this expression yields:

a_9 = 11 * (6561/65536) ≈ 1.10

Therefore, the 9th term of the geometric sequence a_n is approximately 1.10.

Alternative Approach

While the direct substitution method provides a straightforward solution, let's explore an alternative approach to further solidify our understanding. We can leverage the recursive nature of geometric sequences to arrive at the 9th term. Recall that each term is obtained by multiplying the previous term by the common ratio.

Starting with the first term, a_1 = 11, we can iteratively calculate the subsequent terms by repeatedly multiplying by the common ratio, 3/4:

  • a_2 = a_1 * (3/4) = 11 * (3/4) = 8.25
  • a_3 = a_2 * (3/4) = 8.25 * (3/4) = 6.1875
  • a_4 = a_3 * (3/4) = 6.1875 * (3/4) = 4.640625
  • a_5 = a_4 * (3/4) = 4.640625 * (3/4) = 3.48046875
  • a_6 = a_5 * (3/4) = 3.48046875 * (3/4) = 2.6103515625
  • a_7 = a_6 * (3/4) = 2.6103515625 * (3/4) = 1.957763671875
  • a_8 = a_7 * (3/4) = 1.957763671875 * (3/4) = 1.46832275390625
  • a_9 = a_8 * (3/4) = 1.46832275390625 * (3/4) ≈ 1.10

This iterative approach, while more computationally intensive, provides a step-by-step understanding of how the sequence progresses towards the 9th term. It reaffirms our earlier result, confirming that a_9 is indeed approximately 1.10.

Conclusion

In this comprehensive exploration, we successfully combined the functions f(n) = 11 and g(n) = (3/4)^(n-1) to construct a geometric sequence, a_n. By recognizing the constant nature of f(n) and the geometric essence of g(n), we formulated the sequence as a_n = 11 * (3/4)^(n-1). We then employed both direct substitution and an iterative approach to calculate the 9th term, arriving at the consistent result of approximately 1.10.

This endeavor highlights the power of combining mathematical functions to create new sequences with desired properties. The geometric sequence, with its constant ratio and predictable pattern, serves as a fundamental concept in mathematics and finds applications in diverse fields, ranging from finance to physics.

By mastering the principles of geometric sequences and function manipulation, we equip ourselves with valuable tools for tackling a wide array of mathematical challenges.