Introduction to Geometric Sequences and Exponential Relationships

The sequence S 1 2 4 8 16... is a fascinating example of a geometric sequence, where each term is a power of 2. This sequence demonstrates a fundamental mathematical pattern and an exponential relationship. In this article, we will explore how to express this sequence in equation form and understand its underlying principles.

Understanding the Sequence

The sequence S 1, 2, 4, 8, 16... can be recognized as an exponential pattern where each term is a power of 2. Let's break down the sequence to understand its structure.

Expressing the Sequence in Equation Form

The sequence can be expressed mathematically using the formula:

(S_n 2^{n-1})

Here, n represents the position of the term in the sequence, starting from n 1. Let's illustrate this with a few examples:

For n 1:

(S_1 2^{1-1} 2^0 1)

For n 2:

(S_2 2^{2-1} 2^1 2)

For n 3:

(S_3 2^{3-1} 2^2 4)

For n 4:

(S_4 2^{4-1} 2^3 8)

For n 5:

(S_5 2^{5-1} 2^4 16)

This formula effectively captures the pattern where each term is a power of 2. By adjusting the exponent, we can generate each subsequent term in the sequence.

Alternative Expressions

An alternative expression for the sequence can be written as:

(a_n 2^n)

Here, a is the constant base (2 in this case), and n represents the position of the term in the sequence, starting from the second term (n 2). Let's verify this with the first few terms:

For n 2:

(a_2 2^2 4)

For n 3:

(a_3 2^3 8)

For n 4:

(a_4 2^4 16)

And so on...

Geometric Sequence Formula and Generalization

A geometric sequence is defined by the general formula:

(a_n ar^{n-1})

Where:

(a) is the first term of the sequence. (r) is the common ratio of the sequence. (n) is the position of the term in the sequence.

For the sequence S 1, 2, 4, 8, 16..., let's determine the values of a and r:

The first term a 1. The common ratio r can be found by dividing any term by its preceding term:

(r 2/1 2)

Inserting these values into the general formula, we get:

(a_n 1 cdot 2^{n-1} 2^{n-1})

Conclusion

In conclusion, the sequence S 1, 2, 4, 8, 16... can be succinctly expressed in equation form using the formula (S_n 2^{n-1}). This formula effectively captures the exponential relationship inherent in the sequence, allowing us to predict and generate any term in the sequence with ease.