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.