Find a formula for the general term a^n of the sequence assuming the pattern of the first few terms continues. [10/4, 10/16, 10/64, 10/256, 10/1024]

The sequence you provided is:

\[
a_1 = \frac{10}{4}, \quad a_2 = \frac{10}{16}, \quad a_3 = \frac{10}{64}, \quad a_4 = \frac{10}{256}, \quad a_5 = \frac{10}{1024}
\]

We notice that the denominators follow a pattern:

\[
4, 16, 64, 256, 1024
\]

These are powers of 4:
\[
4 = 4^1, \quad 16 = 4^2, \quad 64 = 4^3, \quad 256 = 4^4, \quad 1024 = 4^5
\]

Thus, the general term for the denominator is \( 4^n \).

The numerators are consistently 10.

Thus, the general term \( a_n \) for the sequence can be expressed as:

\[
a_n = \frac{10}{4^n}
\]

This formula applies for all terms in the sequence.

Would you like more details or have any further questions? Here are 5 related questions that might expand your understanding:

1. How do powers of numbers behave in different types of sequences?
2. Can you derive the formula for geometric sequences?
3. What are common differences and ratios in arithmetic and geometric sequences?
4. How can you find the sum of an infinite geometric series?
5. How does the growth rate of this sequence compare to an exponential sequence?

**Tip**: When analyzing sequences, identifying patterns in the numerators and denominators separately can help in deriving a formula!

Explore how to find the general term of a geometric sequence where the numerators are constant (10) and the denominators follow powers of 4. Learn to derive the formula a_n = 10 / 4^n, suitable for students in Grades 9-11, covering concepts like exponentiation and sequence patterns.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation