Answer:
[tex]\textsf{19.} \quad S_{20}=1490[/tex]
33. None of these are correct.
Step-by-step explanation:
Question 19
General form of an arithmetic sequence:
[tex]a_n=a+(n-1)d[/tex]
where:
- [tex]a_n[/tex] is the nth term
- a is the first term
- d is the common difference between terms
Given values:
- [tex]a_n[/tex] = 27
- n = 20
- d = -5
Substitute the given values into the formula and solve for a:
[tex]\implies 27=a+(20-1)(-5)[/tex]
[tex]\implies 27=a+(19)(-5)[/tex]
[tex]\implies 27=a-95[/tex]
[tex]\implies a=27+95[/tex]
[tex]\implies a=122[/tex]
Sum of the first n terms of an arithmetic series:
[tex]S_n=\dfrac12n[2a+(n-1)d][/tex]
where:
- a is the first term.
- d is the common difference.
Given values:
Substitute the given values into the formula and solve:
[tex]\implies S_{20}=\dfrac{1}{2}(20)[2(122)+(20-1)(-5)][/tex]
[tex]\implies S_{20}=10[244-95][/tex]
[tex]\implies S_{20}=10[149][/tex]
[tex]\implies S_{20}=1490[/tex]
Question 33
Sum of the first n terms of a geometric series:
[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
where:
- a is the first term
- r is the common ratio
Given values:
Substitute the given values into the formula and solve:
[tex]\implies S_6=\dfrac{1458\left(1-\frac{1}{3}^6\right)}{1-\frac{1}{3}}[/tex]
[tex]\implies S_6=\dfrac{1456}{\frac{2}{3}}[/tex]
[tex]\implies S_6=\dfrac{1456 \cdot 3}{2}[/tex]
[tex]\implies S_6=2184[/tex]
Learn more about arithmetic sequences here:
https://brainly.com/question/27953040
Learn more about geometric series here:
https://brainly.com/question/27948054
