Evaluate the sum of each of the following series: (1) $$ 7+\frac{15}{2}+8+\frac{17}{2}+\ldots $$ (to 15 terms) (2) $$ 2-6+18-54+\ldots-39366 $$ b wonke (3) $$ 100+50+25+\frac{25}{2}+\ldots $$ (to infinity) (4) $$ 6+2-2-6-\ldots-30 $$ (5) $$ 2-1+7-5+12-9+17-13+\ldots+72-57 $$ (6) $$ 3 x^{2}+9 x^{3}+27 x^{4}+\ldots+6561 x $$

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Find the characteristics of these numbers by following steps:
- step0: Find the mean:
  $$3,9,27,6561$$
- step1: Write the expression of mean:
  $$\textrm{mean}=\frac{3+9+27+6561}{4}$$
- step2: Calculate:
  $$\textrm{mean}=\frac{6600}{4}$$
- step3: Calculate:
  $$\textrm{mean}=1650$$
Find the arithmetic mean of $$ 2,-6,18,-54,-39366 $$
Find the characteristics of these numbers by following steps:
- step0: Find the mean:
  $$2,-6,18,-54,-39366$$
- step1: Write the expression of mean:
  $$\textrm{mean}=\frac{2-6+18-54-39366}{5}$$
- step2: Calculate:
  $$\textrm{mean}=\frac{-39406}{5}$$
- step3: Calculate:
  $$\textrm{mean}=-7881.2$$
Find the arithmetic mean of $$ 6,2,-2,-6,-30 $$
Find the characteristics of these numbers by following steps:
- step0: Find the mean:
  $$6,2,-2,-6,-30$$
- step1: Write the expression of mean:
  $$\textrm{mean}=\frac{6+2-2-6-30}{5}$$
- step2: Calculate:
  $$\textrm{mean}=\frac{-30}{5}$$
- step3: Calculate:
  $$\textrm{mean}=-6$$
Find the arithmetic mean of $$ 2,-1,7,-5,12,-9,17,-13,72,-57 $$
Find the characteristics of these numbers by following steps:
- step0: Find the mean:
  $$2,-1,7,-5,12,-9,17,-13,72,-57$$
- step1: Write the expression of mean:
  $$\textrm{mean}=\frac{2-1+7-5+12-9+17-13+72-57}{10}$$
- step2: Calculate:
  $$\textrm{mean}=\frac{25}{10}$$
- step3: Calculate:
  $$\textrm{mean}=2.5$$
Find the arithmetic mean of $$ 100,50,25,25/2 $$
Find the characteristics of these numbers by following steps:
- step0: Find the mean:
  $$100,50,25,\frac{25}{2}$$
- step1: Write the expression of mean:
  $$\textrm{mean}=\frac{100+50+25+\frac{25}{2}}{4}$$
- step2: Calculate:
  $$\textrm{mean}=\frac{\frac{375}{2}}{4}$$
- step3: Calculate:
  $$\textrm{mean}=46.875$$
Find the arithmetic mean of $$ 7,15/2,8,17/2 $$
Find the characteristics of these numbers by following steps:
- step0: Find the mean:
  $$7,\frac{15}{2},8,\frac{17}{2}$$
- step1: Write the expression of mean:
  $$\textrm{mean}=\frac{7+\frac{15}{2}+8+\frac{17}{2}}{4}$$
- step2: Calculate:
  $$\textrm{mean}=\frac{31}{4}$$
- step3: Calculate:
  $$\textrm{mean}=7.75$$
Let's evaluate each of the series step by step:

### (1) Series: $$ 7+\frac{15}{2}+8+\frac{17}{2}+\ldots $$ (to 15 terms)

This series alternates between integers and fractions. The pattern can be observed as follows:
- The integer part: $$ 7, 8, 9, \ldots $$ (which is an arithmetic sequence starting from 7)
- The fractional part: $$ \frac{15}{2}, \frac{17}{2}, \ldots $$ (which is also an arithmetic sequence starting from $$ \frac{15}{2} $$)

The first 15 terms consist of 8 integers and 7 fractions.

Calculating the sum:
- Sum of integers: $$ 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 70 $$
- Sum of fractions: $$ \frac{15}{2} + \frac{17}{2} + \frac{19}{2} + \frac{21}{2} + \frac{23}{2} + \frac{25}{2} + \frac{27}{2} = \frac{15 + 17 + 19 + 21 + 23 + 25 + 27}{2} = \frac{147}{2} = 73.5 $$

Total sum = $$ 70 + 73.5 = 143.5 $$

### (2) Series: $$ 2-6+18-54+\ldots-39366 $$

This series is a geometric series where:
- First term $$ a = 2 $$
- Common ratio $$ r = -3 $$

To find the number of terms, we can use the formula for the $$ n $$-th term of a geometric series:
$$
a_n = a \cdot r^{n-1}
$$
Setting $$ a_n = -39366 $$:
$$
-39366 = 2 \cdot (-3)^{n-1}
$$
Solving for $$ n $$:
$$
-19683 = (-3)^{n-1}
$$
This gives $$ n-1 = 9 $$ or $$ n = 10 $$.

The sum of the first $$ n $$ terms of a geometric series is given by:
$$
S_n = a \frac{1 - r^n}{1 - r}
$$
Substituting the values:
$$
S_{10} = 2 \frac{1 - (-3)^{10}}{1 - (-3)} = 2 \frac{1 - 59049}{4} = 2 \cdot \frac{-59048}{4} = -29524
$$

### (3) Series: $$ 100+50+25+\frac{25}{2}+\ldots $$ (to infinity)

This series is a geometric series where:
- First term $$ a = 100 $$
- Common ratio $$ r = \frac{1}{2} $$

The sum of an infinite geometric series is given by:
$$
S = \frac{a}{1 - r}
$$
Substituting the values:
$$
S = \frac{100}{1 - \frac{1}{2}} = \frac{100}{\frac{1}{2}} = 200
$$

### (4) Series: $$ 6+2-2-6-\ldots-30 $$

This series alternates between positive and negative terms. The pattern can be observed as:
- Positive terms: $$ 6, 2 $$
- Negative terms: $$ -2, -6, -10, -14, -18, -22, -26, -30 $$

Calculating the sum:
- Positive terms: $$ 6 + 2 = 8 $$
- Negative terms: $$ -2 - 6 - 10 - 14 - 18 - 22 - 26 - 30 = -108 $$

Total sum = $$ 8 - 108 = -100 $$

### (5) Series: $$ 2-1+7-5+12-9+17-13+\ldots+72-57 $$

This series can be split into two parts:
- Positive terms: $$ 2, 7, 12, 17, \ldots, 72 $$ (an arithmetic sequence with first term 2 and common difference 5)
- Negative terms: $$ -1, -5, -9, -13, \ldots, -57 $$ (an arithmetic sequence with first term -1 and common difference -4)

Calculating the sum:
- Positive terms: The last term can be found using $$ a_n = a + (n-1)d $$:
$$
72 = 2 + (n-1)5 \implies 70 = (n-1)5 \implies n-1 = 14 \implies n = 15
$$
Sum of positive terms:
$$
S_{15} = \frac{15}{2} (2 + 72) = \frac{15}{2} \cdot 74 = 555
$$

- Negative terms: The last term can be found similarly:
$$
-57 = -1 + (n-1)(-4) \implies -56 = (n-1)(-4) \implies n-1 = 14 \implies n = 15
$$
Sum of negative terms:
$$
S_{15} = \frac{15}{2} (-1 - 57) = \frac{15}{2} \cdot (-58) = -435
$$

Total sum = $$ 555 - 435 = 120 $$

### (6) Series: $$ 3 x^{2}+9 x^{3}+27 x^{4}+\ldots+6561 x $$

This series is a geometric series where:
- First term $$ a = 3x^2 $$
- Common ratio $$ r = 3x $$

To find the number of terms, we can set the last term equal to $$ 6561x $$:
$$
3x^2 (3x)^{n-1} = 6561x
$$
Dividing both sides by $$ x $$ (assuming $$ x \neq 0 $$):
$$
3x^{n-1} = 6561
$$
Solving for $$ n $$:
$$
x^{n-1} = \frac{6561}{3} = 2187
$$
Since $$ 2187 = 3^7 $$, we have $$ n-1 = 7 $$ or $$ n = 8 $$.

The sum of the first $$ n $$ terms of a geometric series is given by:
\[
S_n = a
(1) The sum of the series $$ 7+\frac{15}{2}+8+\frac{17}{2}+\ldots $$ (to 15 terms) is **143.5**. (2) The sum of the series $$ 2-6+18-54+\ldots-39366 $$ is **-29524**. (3) The sum of the infinite series $$ 100+50+25+\frac{25}{2}+\ldots $$ is **200**. (4) The sum of the series $$ 6+2-2-6-\ldots-30 $$ is **-100**. (5) The sum of the series $$ 2-1+7-5+12-9+17-13+\ldots+72-57 $$ is **120**. (6) The sum of the series $$ 3x^{2}+9x^{3}+27x^{4}+\ldots+6561x $$ is $$ \frac{3x^{2}(3x)^{7} - 3x^{2}}{1 - 3x} $$.

Evaluate the sum of each of the following series:
(1) (to 15 terms)
(2)
b wonke
(3) (to infinity)
(4)
(5)
(6)

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Evaluate the sum of each of the following series: (1) (to 15 terms) (2) b wonke (3) (to infinity) (4) (5) (6)