Let's solve each question step-by-step:
### 5. Sum [tex]\( S_8 \)[/tex] for the General Term [tex]\( a_n = 3 - 5n \)[/tex]
To find the sum [tex]\( S_8 \)[/tex] of the first 8 terms of the sequence, we first need to determine each term from [tex]\( a_1 \)[/tex] to [tex]\( a_8 \)[/tex]:
[tex]\[
\begin{align*}
a_1 &= 3 - 5(1) = 3 - 5 = -2 \\
a_2 &= 3 - 5(2) = 3 - 10 = -7 \\
a_3 &= 3 - 5(3) = 3 - 15 = -12 \\
a_4 &= 3 - 5(4) = 3 - 20 = -17 \\
a_5 &= 3 - 5(5) = 3 - 25 = -22 \\
a_6 &= 3 - 5(6) = 3 - 30 = -27 \\
a_7 &= 3 - 5(7) = 3 - 35 = -32 \\
a_8 &= 3 - 5(8) = 3 - 40 = -37 \\
\end{align*}
\][/tex]
Now, sum these terms:
[tex]\[
S_8 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8
\][/tex]
[tex]\[
S_8 = (-2) + (-7) + (-12) + (-17) + (-22) + (-27) + (-32) + (-37)
\][/tex]
[tex]\[
S_8 = -2 - 7 - 12 - 17 - 22 - 27 - 32 - 37
\][/tex]
[tex]\[
S_8 = -156
\][/tex]
So the answer is:
A. -156
### 6. Sum [tex]\( \sum_{k=1}^{\infty} 5^{3-k} \)[/tex]
Recognize that this is an infinite geometric series. The first term [tex]\( a \)[/tex] and the common ratio [tex]\( r \)[/tex] need to be identified:
[tex]\[
a = 5^{3-1} = 5^2 = 25
\][/tex]
The common ratio [tex]\( r \)[/tex] is:
[tex]\[
r = 5^{3-(k+1)} / 5^{3-k} = 5^{-1} = \frac{1}{5}
\][/tex]
The sum [tex]\( S \)[/tex] of an infinite geometric series [tex]\( \sum_{k=0}^{\infty} ar^k \)[/tex] is given by:
[tex]\[
S = \frac{a}{1 - r}
\][/tex]
Substitute [tex]\( a \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[
S = \frac{25}{1 - \frac{1}{5}} = \frac{25}{\frac{4}{5}} = 25 \times \frac{5}{4} = \frac{125}{4}
\][/tex]
So the answer is:
C. [tex]\( \frac{125}{4} \)[/tex]
### 7. Which one is a convergent sequence as [tex]\( n \)[/tex] approaches infinity?
Let's examine each sequence:
- [tex]\( (\sqrt{2})^n \)[/tex]: Since [tex]\( \sqrt{2} > 1 \)[/tex], this sequence will grow exponentially without bound and hence diverge.
- [tex]\( \left(\frac{2}{\sqrt{3}}\right)^n \)[/tex]: Since [tex]\( \frac{2}{\sqrt{3}} > 1 \)[/tex], this sequence will also grow exponentially without bound and hence diverge.
- [tex]\( -\left(3^n\right) \)[/tex]: Because the base 3 is greater than 1, this sequence will grow without bound in negative direction and hence diverge.
- [tex]\( \left(\frac{4}{5}\right)^n \)[/tex]: Since [tex]\( \frac{4}{5} < 1 \)[/tex], this sequence will decay towards 0 and hence is convergent.
So the answer is:
D. [tex]\( \left(\frac{4}{5}\right)^n \)[/tex]
