The problem in the image asks us to find the sum of the first 20 terms of the sequence $1.5, 1.45, 1.40, 1.35, \dots$.

Step 1: Identify the sequence type

This sequence is arithmetic because the difference between consecutive terms is constant: $\text{Common difference, } d = 1.45 - 1.5 = -0.05.$

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

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

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

Step 3: Substituting values

Here:

• $a = 1.5$,
• $d = -0.05$,
• $n = 20$.

Substitute these values into the formula: $S_{20} = \frac{20}{2} \left( 2(1.5) + (20-1)(-0.05) \right).$

Simplify step-by-step: $S_{20} = 10 \left( 3 + 19(-0.05) \right),$ $S_{20} = 10 \left( 3 - 0.95 \right),$ $S_{20} = 10 \times 2.05 = 20.5.$

Final Answer:

The sum of the first 20 terms is: $\boxed{20.5}.$

Would you like further clarification or a detailed breakdown of any step?

Follow-up questions to explore:

1. What is the formula for the nth term of an arithmetic sequence?
2. How do we recognize an arithmetic sequence from a list of terms?
3. What happens if the common difference is positive rather than negative?
4. How would you compute the sum if $n$ were very large?
5. Can you derive the sum formula for arithmetic sequences from scratch?

Tip:

When solving sequence problems, always verify the type of sequence (arithmetic, geometric, etc.) first to apply the correct formula!