Question

example 9
find the sum of each geometric series.

1. $sum_{k=1}^{7} 4(-3)^{k - 1}$
2. $sum_{k=1}^{8} (-3)(-2)^{k - 1}$
3. $sum_{k=1}^{9} (-1)(4)^{k - 1}$
4. $sum_{k=1}^{10} 5(-1)^{k - 1}$

Explanation:

Problem 34:

Step1: Identify geom series parameters

For $\sum_{k=1}^{7} 4(-3)^{k-1}$, first term $a=4$, ratio $r=-3$, $n=7$.

Step2: Apply finite geom sum formula

Sum formula: $S_n = a\frac{1-r^n}{1-r}$
$S_7 = 4\cdot\frac{1-(-3)^7}{1-(-3)}$

Step3: Calculate powers and simplify

$(-3)^7=-2187$, so $1-(-2187)=2188$; $1-(-3)=4$
$S_7 = 4\cdot\frac{2188}{4}=2188$

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Problem 35:

Step1: Identify geom series parameters

For $\sum_{k=1}^{8} (-3)(-2)^{k-1}$, first term $a=-3$, ratio $r=-2$, $n=8$.

Step2: Apply finite geom sum formula

$S_n = a\frac{1-r^n}{1-r}$
$S_8 = -3\cdot\frac{1-(-2)^8}{1-(-2)}$

Step3: Calculate powers and simplify

$(-2)^8=256$, so $1-256=-255$; $1-(-2)=3$
$S_8 = -3\cdot\frac{-255}{3}=255$

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Problem 36:

Step1: Identify geom series parameters

For $\sum_{k=1}^{9} (-1)(4)^{k-1}$, first term $a=-1$, ratio $r=4$, $n=9$.

Step2: Apply finite geom sum formula

$S_n = a\frac{1-r^n}{1-r}$
$S_9 = -1\cdot\frac{1-4^9}{1-4}$

Step3: Calculate powers and simplify

$4^9=262144$, so $1-262144=-262143$; $1-4=-3$
$S_9 = -1\cdot\frac{-262143}{-3}=-87381$

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Problem 37:

Step1: Identify geom series parameters

For $\sum_{k=1}^{10} 5(-1)^{k-1}$, first term $a=5$, ratio $r=-1$, $n=10$.

Step2: Apply finite geom sum formula

$S_n = a\frac{1-r^n}{1-r}$
$S_{10} = 5\cdot\frac{1-(-1)^{10}}{1-(-1)}$

Step3: Calculate powers and simplify

$(-1)^{10}=1$, so $1-1=0$; $1-(-1)=2$
$S_{10} = 5\cdot\frac{0}{2}=0$

Answer:

1. $\boldsymbol{2188}$
2. $\boldsymbol{255}$
3. $\boldsymbol{-87381}$
4. $\boldsymbol{0}$