https://file.aiquickdraw.com/mathbot/file/17272663401635a18tp2f.jpg

The problem describes a sequence where the nth term is given by the expression \( a^{3n} - 3 \). We are told that the second term of the sequence is 61, and we need to determine which of the given values of \( a \) could be correct.

### Step-by-Step Solution:

The general term for the sequence is:

\[
T_n = a^{3n} - 3
\]

The second term (\( n = 2 \)) is given as 61. Therefore, we substitute \( n = 2 \) into the equation for the second term:

\[
T_2 = a^{3(2)} - 3 = 61
\]

Simplifying:

\[
a^6 - 3 = 61
\]

\[
a^6 = 64
\]

Now, solve for \( a \):

\[
a = \sqrt[6]{64}
\]

Since \( 64 = 2^6 \), we get:

\[
a = \pm 2
\]

Thus, \( a \) could be either \( 2 \) or \( -2 \).

### Evaluate the given choices:

- \( a = -2 \) → This is listed as option I.
- \( a = 2 \) → This is listed as option II.
- \( a = 4 \) → \( a^6 = 4096 \), so this is **not** a valid option.

### Conclusion:

The correct values of \( a \) are \( -2 \) and \( 2 \), which correspond to options **I or II only**.

The correct answer is:

\[
\boxed{I \text{ or } II \text{ only}}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. What are the general steps to solve an exponential equation like \( a^6 = 64 \)?
2. How can we handle negative exponents in similar sequence problems?
3. What would happen if the sequence were given by \( a^{2n} - 5 \) instead?
4. How does the growth of sequences change depending on the base \( a \) being positive or negative?
5. How would the problem change if we were solving for the third term instead of the second?

**Tip:** Always double-check your algebraic manipulation when solving for powers and roots, especially when working with sequences and series.

The value of the nth term of a sequence is given by the expression a^(3n) - 3. If the second term of the sequence is 61, which of the following could be the value of a? I. -2 II. 2 III. 4

Learn how to solve for the value of a in the sequence a^(3n) - 3, given that the second term equals 61. A detailed step-by-step breakdown is provided for students in Grades 9-12, covering key concepts like sequences, exponents, and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation