To find the sum of the first 20 terms of the given arithmetic sequence:
1. Identify the initial term (first term) and the common difference:
- The first term ([tex]\(a\)[/tex]) of the sequence is 1.5.
2. Calculate the common difference ([tex]\(d\)[/tex]):
- The sequence progresses by subtracting a constant value from each term to get the next term in the sequence.
- The second term is 1.45.
- Therefore, the common difference [tex]\(d\)[/tex] is given by:
[tex]\[
d = \text{{second term}} - \text{{first term}} = 1.45 - 1.5
\][/tex]
[tex]\[
d = -0.05
\][/tex]
3. Determine the number of terms ([tex]\(n\)[/tex]):
- We are asked to find the sum of the first 20 terms, so [tex]\(n = 20\)[/tex].
4. Use the formula for sum of the first [tex]\(n\)[/tex] terms of an arithmetic series:
[tex]\[
S_n = \frac{n}{2} [2a + (n-1)d]
\][/tex]
- Here, [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms.
- Substituting the known values into the formula:
[tex]\[
a = 1.5, \quad d = -0.05, \quad n = 20
\][/tex]
[tex]\[
S_{20} = \frac{20}{2} [2(1.5) + (20-1)(-0.05)]
\][/tex]
[tex]\[
S_{20} = 10 [3 + 19(-0.05)]
\][/tex]
5. Simplify inside the brackets:
[tex]\[
3 + 19(-0.05) = 3 - 0.95 = 2.05
\][/tex]
6. Calculate the product:
[tex]\[
S_{20} = 10 \times 2.05
\][/tex]
[tex]\[
S_{20} = 20.50
\][/tex]
So, the common difference is [tex]\(-0.05\)[/tex], and the sum of the first 20 terms of the given arithmetic sequence is [tex]\(20.50\)[/tex].
