To find the sum of the arithmetic sequence \(6 + 15 + 24 + 33 + \ldots + 105\), we can follow these steps:

1. **Identify the sequence parameters:**
   - The first term \(a_1\) is 6.
   - The common difference \(d\) is found by subtracting the first term from the second term:
     \[
     d = 15 - 6 = 9
     \]
   - The last term \(a_n\) is 105.

2. **Determine the number of terms (\(n\)):**
   - Use the formula for the \(n\)-th term of an arithmetic sequence:
     \[
     a_n = a_1 + (n-1) \times d
     \]
     Plugging in the values:
     \[
     105 = 6 + (n-1) \times 9
     \]
     Solve for \(n\):
     \[
     105 - 6 = (n-1) \times 9
     \]
     \[
     99 = (n-1) \times 9
     \]
     \[
     n-1 = \frac{99}{9} = 11
     \]
     \[
     n = 11 + 1 = 12
     \]
     So, there are 12 terms in the sequence.

3. **Calculate the sum (\(S_n\)):**
   - The sum \(S_n\) of an arithmetic sequence is given by:
     \[
     S_n = \frac{n}{2} \times (a_1 + a_n)
     \]
     Substituting the values:
     \[
     S_{12} = \frac{12}{2} \times (6 + 105) = 6 \times 111 = 666
     \]

So, the sum of the sequence \(6 + 15 + 24 + 33 + \ldots + 105\) is **666**.

Would you like more details or have any questions?

Here are five related questions:
1. How do you find the sum of a geometric sequence?
2. What is the sum of the arithmetic sequence \(2 + 10 + 18 + \ldots + 98\)?
3. How can you determine the sum of the first \(n\) terms of an arithmetic series if the last term is unknown?
4. What is the difference between the sum of the sequence \(6 + 15 + 24 + \ldots + 105\) and the sum of a similar sequence starting at 7 with the same difference?
5. How would you calculate the sum of the squares of the terms in this sequence?

**Tip:** Always identify the common difference and the number of terms in an arithmetic sequence before trying to find the sum. This ensures you use the correct formula and avoid mistakes.

Learn how to calculate the sum of the arithmetic sequence 6 + 15 + 24 + 33 + ... + 105 with a step-by-step solution. This problem involves identifying the sequence parameters, determining the number of terms, and applying the arithmetic sequence sum formula. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation