Introduction to Geometric Sequences

A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number is known as a geometric sequence. In the given sequence, 4, 8, 16, 32, 64, each subsequent term is obtained by multiplying the previous term by 2.

Understanding the Explicit Formula

The explicit formula for a geometric sequence can be written as:

an a1 middot; rn-1

Here, a1 is the first term of the sequence, r is the common ratio, and n is the term number. For the sequence provided, the first term a1 is 4 and the common ratio r is 2.

Step-by-Step Derivation of the Explicit Formula

To find the explicit formula, we simply substitute the known values:

an 4 middot; 2n-1

Finding the 30th Term

To find the 30th term, we substitute n 30 into the formula:

a30 4 middot; 229

Let's calculate 229:

229 536,870,912

Now we multiply this by 4:

a30 4 middot; 536,870,912 2,147,483,648

General Formula and Proof

The sequence can also be expressed as:

an 2n-1

This can be derived as follows:

Each term in the sequence is a power of 2, starting from 22, 23, 24, etc. Hence, the general term can be rewritten as:

2 * 2(n-1) 21 * 2(n-1) 2(1 (n-1)) 2n

Thus, the explicit formula simplifies to:

an 2n-1

For the 30th term, substituting n 30:

a30 229 536,870,912

Finally, multiplying by 4 to get the term:

2,147,483,648

Summary

The explicit formula for the sequence is:

an 2n-1

The 30th term of the sequence is:

2,147,483,648

This confirms our previous calculations and provides a concise way to determine the value of any term in the sequence.

Additional Considerations

Understanding geometric sequences and their explicit formulas is crucial for a wide range of applications in mathematics, science, and technology. Whether you're dealing with exponential growth, decay, or other related concepts, the knowledge of geometric sequences can be very handy.

If you have any further questions or need more detailed explanations, feel free to ask!