Let's analyze the questions from the image step by step and solve them.

Question 9

The sum $S = 100 + 101 + 102 + \dots + 140$.

This is an arithmetic series with:

• First term $a = 100$,
• Last term $l = 140$,
• Number of terms $n$.

We use the formula for the sum of an arithmetic sequence: $S = \frac{n}{2} \times (a + l)$

Step 1: Find the number of terms

The number of terms in an arithmetic sequence is: $n = \frac{l - a}{d} + 1$ where $d$ is the common difference (here $d = 1$).

$n = \frac{140 - 100}{1} + 1 = 41$

Step 2: Calculate the sum

$S = \frac{41}{2} \times (100 + 140) = 20.5 \times 240 = 4920$

So, the correct answer is: $\boxed{a) \ 4920}$

Question 10

The sequence $(U_n)_{n \geq 0}$ is a geometric progression with:

• $U_{20} = 45$,
• $U_{23} = 5625$.

We are tasked to find the sum: $S = U_{18} + U_{19} + \dots + U_{28}.$

Step 1: Determine the common ratio ($r$)

In a geometric sequence, each term is multiplied by the common ratio $r$. Using the formula for the $n$-th term: $U_{23} = U_{20} \cdot r^{23-20} = U_{20} \cdot r^3$ $5625 = 45 \cdot r^3$ $r^3 = \frac{5625}{45} = 125 \quad \Rightarrow \quad r = \sqrt[3]{125} = 5$

Step 2: Express $U_{18}$ and the terms

From $U_{20} = 45$ and $r = 5$, we can move backward to find $U_{18}$: $U_{20} = U_{18} \cdot r^2 \quad \Rightarrow \quad U_{18} = \frac{U_{20}}{r^2} = \frac{45}{5^2} = \frac{45}{25} = 1.8$

Step 3: Find the sum of terms

The sum of a finite geometric series is given by: $S = U_{18} \cdot \frac{1 - r^{n}}{1 - r}$ where $n$ is the number of terms. From $U_{18}$ to $U_{28}$, there are: $n = 28 - 18 + 1 = 11 \ \text{terms.}$

Substitute: $S = 1.8 \cdot \frac{1 - 5^{11}}{1 - 5}$ $S = 1.8 \cdot \frac{1 - 48828125}{-4}$ $S = 1.8 \cdot \frac{-48828124}{-4}$ $S = 1.8 \cdot 12207031 = 21972655.8$

Approximating, the correct answer is: $\boxed{c) \ 21 \, 972 \, 656}$

Final Answers:

• Question 9: $\boxed{a \ 4920}$,
• Question 10: $\boxed{c \ 21 \, 972 \, 656}$.

Would you like a more detailed explanation or further breakdown?

Related Questions:

1. What is the formula for the sum of an arithmetic sequence?
2. How do you determine the common ratio of a geometric sequence?
3. What are the conditions for using the geometric series sum formula?
4. Can you derive the sum formula for an arithmetic series?
5. How does changing the number of terms in the sequence affect the sum?

Tip:

Always double-check calculations in geometric sequences, especially powers of the common ratio, as they grow very quickly!