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A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence can make numbers grow very quickly or shrink rapidly, depending on whether the common ratio is greater than or less than 1. It's a pattern found in many mathematical and real-world scenarios.

For example, if we start with the number 2 and have a common ratio of 3, the sequence would be: 2, 6, 18, 54, and so on. Each term is the previous term multiplied by 3.

In contrast to arithmetic sequences, where you add a fixed number to each term to get the next, in geometric sequences you multiply by a fixed ratio. This difference creates quite distinct patterns.

• Each term is multiplied by the common ratio.
• The terms can increase or decrease exponentially.
• Used in things like calculating compound interest, population growth, etc.

The sequence featuring in the exercise where each term is of the form \(2^{n+2}\) follows a geometric progression because our tests confirm that the ratio between consecutive terms is constant, as shown by \(\frac{16}{8} = \frac{32}{16} = 2\).

The common ratio is a key element that makes a sequence geometric. This constant is the multiplier applied to each term to obtain the next. Identifying the common ratio confirms the nature of the sequence and allows for predictions about future terms.

For the sequence given by \(b_n = 2^{n+2}\), we found our common ratio by dividing consecutive terms. So, from calculating \(\frac{b_2}{b_1} = \frac{16}{8} = 2\) and \(\frac{b_3}{b_2} = \frac{32}{16} = 2\), we confirm that the ratio is consistently 2.

Knowing this is crucial as it gives insight into the sequence's behavior:

• A ratio greater than 1 means each term is greater than the previous, indicating growth.
• A ratio less than 1 shows a decrease, so terms shrink.
• Predicting future terms becomes systematic using the formula \(b_n = b_1 \, r^{(n-1)}\).

The clear consistency of the ratio confirms the productivity of a geometric sequence by giving a predictable progression of terms.

Calculating any term in a geometric sequence involves a straightforward method using the identified common ratio and the first term of the sequence. The formula used is straightforward: \(b_n = b_1 \, r^{(n-1)}\). This formula allows us to find any term without needing to know all the preceding ones.

Steps For Calculation:

• Identify the first term, \(b_1\).
• Determine the common ratio, \(r\).
• Substitute \( n \) for the term you want to find.

Let's use the formula to calculate the 10th term for our sequence \(b_n = 2^{n+2}\). We already know \(b_1 = 8\) and \(r = 2\). Substituting these into the formula gives:
\[ b_{10} = 8 \cdot 2^{(10-1)} = 8 \cdot 2^9 = 8 \cdot 512 = 4096 \]

This computation shows that the 10th term equals 4096, thanks to our systematic approach using the properties of geometric sequences.