write the following series in sigma notation and find their sum.
1+9+45+81

The given series is: 

\[
1 + 9 + 45 + 81
\]

### Step 1: Identify the Pattern
We can observe the pattern in the terms:
- \(1 = 1^2 \times 1\)
- \(9 = 3^2 \times 1\)
- \(45 = 3^2 \times 5\)
- \(81 = 3^4 \times 1\)

But this sequence does not form a simple arithmetic or geometric progression. However, we can see that each term involves powers of 3 and different multipliers. 

### Step 2: Express the Series in Sigma Notation

To express the series in sigma notation, let's analyze it further.

The given series can be rewritten as:

\[
1 + 9 + 45 + 81 = 1 \times 3^0 + 3^2 \times 1 + 3^2 \times 5 + 3^4 \times 1
\]

However, a more systematic approach to write this in sigma notation is:

\[
\sum_{n=1}^{4} (3^{2n - 2} \times c_n)
\]

where \( c_n \) corresponds to the coefficients (1, 1, 5, 1) respectively.

Given the complexity of the pattern in the coefficients, the direct sum cannot be found using a simple formula, but since this is a finite series, the sum is simply:

\[
1 + 9 + 45 + 81 = 136
\]

### Final Answer

The sum of the series \(1 + 9 + 45 + 81\) is:

\[
\sum_{n=1}^{4} (3^{2n - 2} \times c_n) = 136
\]

Would you like more details, or do you have any questions?

### Related Questions
1. How do you express an arithmetic series in sigma notation?
2. How do you express a geometric series in sigma notation?
3. How can you determine the general term of a series?
4. How do you find the sum of an arithmetic series?
5. What are the common patterns in sigma notation series?

### Tip
Identifying the pattern in a sequence is key to writing it in sigma notation. Look for powers, coefficients, and common differences or ratios.

Learn how to express and sum a series in sigma notation, specifically the series 1 + 9 + 45 + 81. This problem involves identifying patterns, coefficients, and applying sigma notation to find the sum of the series. Suitable for advanced high school students and those studying series and sequences in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation