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

To write the series \( 1 + 9 + 45 + 81 \) in sigma notation, let's first observe the pattern in the sequence:

1. The first term is \( 1 \).
2. The second term is \( 9 = 3^2 \).
3. The third term is \( 45 = 5 \times 9 \).
4. The fourth term is \( 81 = 9^2 \).

The sequence can be written as:
\[
1, \, 3^2, \, 5 \times 9, \, 9^2
\]

However, this does not follow a straightforward arithmetic or geometric progression. Let's analyze it in another way by looking for a pattern in the sequence of terms.

Notice:
- \( 1 = 1 \times 1 \)
- \( 9 = 3 \times 3 \)
- \( 45 = 5 \times 9 \)
- \( 81 = 9 \times 9 \)

Now, the series can be represented as:
\[
\sum_{n=1}^{4} (2n-1) \times (2n-1)
\]

Thus, the sigma notation for this series is:
\[
\sum_{n=1}^{4} (2n-1)^2
\]

To find the sum:
\[
\text{Sum} = \sum_{n=1}^{4} (2n-1)^2 = 1^2 + 3^2 + 5^2 + 7^2
\]
\[
= 1 + 9 + 25 + 49 = 84
\]

So, the sum of the series is **84**.

Would you like more details on the steps or any clarifications?

Here are some related questions:

1. How can we identify the type of sequence (arithmetic, geometric, etc.)?
2. What is the general formula for the sum of squares of odd numbers?
3. How can this method be applied to sequences with more terms?
4. Can this approach be extended to sequences that are not quadratic?
5. What are other ways to represent sequences using sigma notation?

**Tip:** Always check if a sequence follows a recognizable pattern, such as arithmetic or geometric, before writing it in sigma notation.

Learn how to write the series 1 + 9 + 45 + 81 in sigma notation and calculate its sum. This problem involves identifying patterns in series, applying sigma notation for sum calculation, and using the formula for the sum of squares of odd numbers. Suitable for high school students studying advanced mathematical sequences.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation