Problem Breakdown:

We are tasked with determining the sum of only the odd values in the given sequence: $11, 22, 33, \dots, 308, 319, 330$

Key Information:

1. The series contains 30 terms.
2. It alternates between odd and even numbers starting from 11.

Steps to Solve:

1. Identify the odd terms in the sequence:

Since the series alternates between odd and even numbers:

• Odd terms are the 1st, 3rd, 5th, ..., 29th terms.
• There are 15 odd terms in total.

The odd terms are: $11, 33, 55, \dots, 319$

2. Confirm the odd subsequence:

This is an arithmetic sequence where:

• The first term $a = 11$,
• The common difference $d = 22$ (difference between two consecutive odd terms).

The $n$-th term of an arithmetic sequence is: $a_n = a + (n-1)d$ For $n = 15$ (last odd term): $a_{15} = 11 + (15-1) \cdot 22 = 11 + 308 = 319$ Thus, the odd subsequence is correctly $11, 33, 55, \dots, 319$.

3. Compute the sum of the odd terms:

The sum of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a + l)$ where:

• $n = 15$ (number of odd terms),
• $a = 11$ (first term),
• $l = 319$ (last term).

Substitute: $S_{15} = \frac{15}{2} \cdot (11 + 319) = \frac{15}{2} \cdot 330 = 15 \cdot 165 = 2475$

Final Answer:

The sum of the odd values in the sequence is: $\boxed{2475}$

Do you want a detailed explanation of any step?

Related Questions:

1. What is the sum of all even values in this sequence?
2. How many terms are needed for the sum of odd terms to exceed 5000?
3. What is the average of the odd numbers in this sequence?
4. How would the sum change if the sequence contained 50 terms instead of 30?
5. Can you derive a formula for the sum of odd terms in a sequence like this?

Tip:

When working with alternating sequences, separate the odd and even terms first to simplify calculations!