4) Find the specified term of the following

a) Identify the 7th term of the sequence: 4, 20, 100, ...

The given sequence is:

\[ 4, 20, 100, \dots \]

We notice that each term is multiplying by 5, so this could be a geometric sequence. Let’s verify that.

To confirm:
- \( \frac{20}{4} = 5 \)
- \( \frac{100}{20} = 5 \)

Hence, this is indeed a geometric sequence with the first term \( a = 4 \) and the common ratio \( r = 5 \).

The general formula for the \(n\)-th term of a geometric sequence is:

\[
T_n = a \cdot r^{n-1}
\]

Where:
- \( T_n \) is the \(n\)-th term,
- \( a \) is the first term,
- \( r \) is the common ratio, and
- \( n \) is the position of the term.

Now, to find the 7th term:

\[
T_7 = 4 \cdot 5^{7-1} = 4 \cdot 5^6
\]

Let’s compute \( 5^6 \):

\[
5^6 = 15625
\]

Now calculate the 7th term:

\[
T_7 = 4 \cdot 15625 = 62500
\]

Thus, the 7th term is:

\[
T_7 = 62500
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What is the 10th term of this geometric sequence?
2. Can you derive the formula for the sum of the first \(n\) terms of a geometric sequence?
3. How does changing the common ratio affect the growth of a geometric sequence?
4. What happens if the common ratio is a fraction between 0 and 1?
5. How do geometric sequences compare to arithmetic sequences in terms of growth?

**Tip**: Always check for the common ratio or difference when identifying a sequence type—geometric or arithmetic.

Identify the 7th term of the sequence: 4, 20, 100, ...

Learn how to calculate the 7th term of the geometric sequence 4, 20, 100 using the formula T_n = a * r^(n-1). Understand the steps involved in determining the common ratio and applying exponents to find T_7 = 62500. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation