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) \( 3 x^{2}+9 x^{3}+27 x^{4}+\ldots+6561 x \)

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