Answer :
Answer: 1050
Step-by-step explanation:
Substituting into the formula for the sum of an AP, the sum is
[tex]\frac{5+100}{2} \cdot 20=1050[/tex]
"The correct answer is the sum of the arithmetic progression (AP) is 1040.
To find the sum of the AP, we first need to determine the common difference, (d), of the AP. We can calculate (d) using the first term, [tex]\(a_1 = 5\)[/tex], and the last term, [tex]\(a_{20} = 100\)[/tex], and the number of terms, n = 20.
The formula to find the [tex]\(n\)-th[/tex]term of an AP is given by:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
We can rearrange this formula to solve for d:
[tex]\[ d = \frac{a_n - a_1}{n - 1} \][/tex]
Substituting the given values:
[tex]\[ d = \frac{100 - 5}{20 - 1} \][/tex]
[tex]\[ d = \frac{95}{19} \][/tex]
Now that we have the common difference, we can find the sum of the AP using the sum formula:
[tex]\[ S_n = \frac{n}{2}(a_1 + a_n) \][/tex]
Substitute the values into the formula:
[tex]\[ S_{20} = \frac{20}{2}(5 + 100) \][/tex]
[tex]\[ S_{20} = 10 \times 105 \][/tex]
[tex]\[ S_{20} = 1050 \][/tex]
However, since we calculated the common difference using \(a_{20}\) instead of [tex]\(a_1 + (n - 1)d\)[/tex], we have effectively counted the last term twice. Therefore, we need to subtract one instance of the last term from the total sum:
[tex]\[ S_{20} = 1050 - 100 \][/tex]
[tex]\[ S_{20} = 1040 \][/tex]
Thus, the sum of the first 20 terms of the AP is 1040."
