The question asks for the sum of the first 10 terms of the sequence: $8, 20, 32, 44, \ldots$. Let's analyze and solve this step by step.

Step 1: Identify the sequence

The sequence appears to be an arithmetic progression (AP) because the difference between consecutive terms is constant: $20 - 8 = 12,\quad 32 - 20 = 12,\quad 44 - 32 = 12.$ So, the common difference $d = 12$, and the first term $a = 8$.

Step 2: Formula for the sum of the first $n$ terms of an AP

The sum of the first $n$ terms of an arithmetic progression is given by: $S_n = \frac{n}{2} \left( 2a + (n-1)d \right),$ where:

• $n$ = number of terms,
• $a$ = first term,
• $d$ = common difference.

Step 3: Plug in the values

For $n = 10$, $a = 8$, and $d = 12$: $S_{10} = \frac{10}{2} \left( 2(8) + (10-1)(12) \right).$

Step 4: Simplify

1. $2a = 2(8) = 16$,
2. $(n-1)d = (10-1)(12) = 9(12) = 108$,
3. $2a + (n-1)d = 16 + 108 = 124$,
4. $S_{10} = \frac{10}{2} \times 124 = 5 \times 124 = 620$.

Final Answer:

The sum of the first 10 terms is: $\boxed{620}$

Would you like further details or clarifications?

Related Questions:

1. What is the general formula for the $n$-th term of this arithmetic sequence?
2. What would the 15th term of this sequence be?
3. How does the sum formula change if the sequence were geometric?
4. What is the sum of the first 20 terms of this sequence?
5. How can we derive the sum formula for an arithmetic progression?

Tip:

When working with arithmetic progressions, always verify the common difference $d$ to confirm that the sequence is consistent before applying formulas.