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

Let's write the given series \(1 + 9 + 45 + 81\) in sigma notation and then find its sum.

### Step 1: Identify the pattern
Observe the series: \(1, 9, 45, 81\).

Let's see if we can express each term in a general form:
1. First term: \(1 = 1 \times 1^2\)
2. Second term: \(9 = 3 \times 3^2\)
3. Third term: \(45 = 5 \times 5^2\)
4. Fourth term: \(81 = 9 \times 9^2\)

The pattern appears to follow the form:
\[
a_n = (2n - 1) \times (2n - 1)^2
\]
Where \(n\) is the term number in the sequence.

### Step 2: Write the series in sigma notation
The series can be written in sigma notation as:
\[
\sum_{n=1}^{4} (2n - 1) \times (2n - 1)^2
\]
This can be simplified to:
\[
\sum_{n=1}^{4} (2n - 1)^3
\]

### Step 3: Calculate the sum of the series
Now, let's calculate the sum by expanding each term:

\[
\text{Sum} = 1^3 + 3^3 + 5^3 + 9^3
\]

Calculate each cube:
\[
1^3 = 1
\]
\[
3^3 = 27
\]
\[
5^3 = 125
\]
\[
9^3 = 729
\]

So, the sum is:
\[
1 + 27 + 125 + 729 = 882
\]

### Final Answer
The series in sigma notation is:
\[
\sum_{n=1}^{4} (2n - 1)^3
\]
The sum of the series is \(882\).

Would you like more details or have any questions? 

Here are five related questions to consider:
1. How would the sigma notation change if the series had 5 terms instead of 4?
2. What is the sum of the series \(1 + 27 + 125 + 343\)?
3. How would you derive a general formula for the sum of a series where each term is a cubic expression?
4. Can you express a similar series using a different polynomial function?
5. How does this series compare to an arithmetic or geometric series in terms of growth rate?

**Tip:** When dealing with series, identifying a pattern or rule governing each term is crucial for expressing the series in sigma notation.

Learn how to write the series 1+9+45+81 in sigma notation and find its sum. Understand the pattern in the series and use the sum of cubes formula to calculate the result. Suitable for high school students studying sequences and series.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation