The sequence given is: 2, 4, 8, 16, ...

Conjecture:

This is a geometric sequence where each term is obtained by multiplying the previous term by 2.

• First term ($a_1$) = 2
• Common ratio ($r$) = 2

General Formula:

The $n$-th term of a geometric sequence can be found using the formula: $a_n = a_1 \times r^{n-1}$ where:

• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the position of the term in the sequence.

Finding the 10th Term:

Substitute $a_1 = 2$, $r = 2$, and $n = 10$ into the formula: $a_{10} = 2 \times 2^{10-1} = 2 \times 2^9 = 2 \times 512 = 1024$

So, the 10th term is 1024.

Explanation of the Pattern:

• Each term is obtained by multiplying the previous term by 2.
• The sequence follows a pattern of powers of 2: $2^1, 2^2, 2^3, 2^4, \ldots$

Do you need more details or have any questions?

Related Questions:

1. What is the 15th term of this sequence?
2. What is the sum of the first 10 terms of this geometric sequence?
3. How can the common ratio affect the growth of a geometric sequence?
4. How does a geometric sequence differ from an arithmetic sequence?
5. Can you write the recursive formula for this geometric sequence?

Tip:

In a geometric sequence, if the common ratio is greater than 1, the terms grow exponentially as $n$ increases!