Question 1 1.1. Debeanine the largest valuc of \( k \) for which: \[ \sum_{n=1}^{k} 2(3)^{n}<30000 \] 1.2. For which values of \( x \) will \( \sum_{k=1}^{\infty}(4 x-1)^{k} \) exisd? Question 2 2.1. Conoider the arithmabic sequence: \( -8 ;-2 ; 4 ; 10 ; \ldots \). 2.1.1. If tho \( n^{\text {th }} \) bam of tes sequence is 148 , defermine bhe value of \( n \). 2.1.2. Calaulate the smallest value of \( n \) for which the sum of the first \( n \) berms of the sequence will be greaber than 10140 (5)

Let's tackle each question step-by-step to ensure clarity and thorough understanding. --- ## **Question 1** ### **1.1. Find the largest value of \( k \) for which:** \[ \sum_{n=1}^{k} 2 \cdot 3^{n} < 30{,}000 \] **Solution:** First, evaluate the sum of the geometric series: \[ S(k) = \sum_{n=1}^{k} 2 \cdot 3^{n} \] Factor out the constant: \[ S(k) = 2 \sum_{n=1}^{k} 3^{n} \] Recall the formula for the sum of a geometric series: \[ \sum_{n=1}^{k} r^{n} = \frac{r^{k+1} - r}{r - 1} \] Applying this formula: \[ S(k) = 2 \left( \frac{3^{k+1} - 3}{3 - 1} \right) = 2 \left( \frac{3^{k+1} - 3}{2} \right) = 3^{k+1} - 3 \] Set the sum less than 30,000: \[ 3^{k+1} - 3 < 30{,}000 \\ 3^{k+1} < 30{,}003 \] Take the logarithm base 3 of both sides to solve for \( k+1 \): \[ k + 1 < \log_{3}(30{,}003) \] Estimate \( \log_{3}(30{,}003) \): \[ 3^{9} = 19{,}683 < 30{,}003 < 59{,}049 = 3^{10} \] \[ \Rightarrow 9 < \log_{3}(30{,}003) < 10 \] \[ \Rightarrow k + 1 < 10 \\ \Rightarrow k < 9 \] Since \( k \) must be an integer, the largest possible value is: \[ k = 8 \] **Answer:** The largest value of \( k \) is **8**. --- ### **1.2. For which values of \( x \) will the series \( \sum_{k=1}^{\infty} (4x - 1)^{k} \) exist?** **Solution:** The given series is a geometric series with: \[ \text{First term} = a = (4x - 1)^1 = 4x - 1 \] \[ \text{Common ratio} = r = 4x - 1 \] A geometric series converges (i.e., exists) if and only if the absolute value of the common ratio is less than 1: \[ |r| < 1 \\ |4x - 1| < 1 \] Solve the inequality: \[ -1 < 4x - 1 < 1 \] Add 1 to all parts: \[ 0 < 4x < 2 \] Divide by 4: \[ 0 < x < \frac{1}{2} \] **Answer:** The series exists for all real numbers \( x \) where \( 0 < x < \dfrac{1}{2} \). --- ## **Question 2** ### **2.1. Consider the arithmetic sequence: \( -8, -2, 4, 10, \ldots \).** First, identify the common difference (\( d \)): \[ d = -2 - (-8) = 6 \] ### **2.1.1. If the \( n^{\text{th}} \) term of the sequence is 148, determine the value of \( n \).** **Solution:** The formula for the \( n^{\text{th}} \) term of an arithmetic sequence is: \[ a_n = a_1 + (n - 1)d \] Given: \[ a_n = 148, \quad a_1 = -8, \quad d = 6 \] Plug in the values: \[ 148 = -8 + (n - 1) \cdot 6 \\ 148 + 8 = 6(n - 1) \\ 156 = 6(n - 1) \\ n - 1 = \frac{156}{6} = 26 \\ n = 26 + 1 = 27 \] **Answer:** The value of \( n \) is **27**. --- ### **2.1.2. Calculate the smallest value of \( n \) for which the sum of the first \( n \) terms of the sequence will be greater than 10,140.** **Solution:** The sum of the first \( n \) terms (\( S_n \)) of an arithmetic sequence is given by: \[ S_n = \frac{n}{2} \left( 2a_1 + (n - 1)d \right) \] Given: \[ a_1 = -8, \quad d = 6 \] Plug in the values: \[ S_n = \frac{n}{2} \left( 2(-8) + (n - 1) \cdot 6 \right) \\ S_n = \frac{n}{2} \left( -16 + 6n - 6 \right) \\ S_n = \frac{n}{2} (6n - 22) \\ S_n = n(3n - 11) \] Set the sum greater than 10,140: \[ n(3n - 11) > 10{,}140 \\ 3n^2 - 11n - 10{,}140 > 0 \] Solve the quadratic inequality \( 3n^2 - 11n - 10{,}140 > 0 \): 1. Find the roots using the quadratic formula: \[ n = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 3 \cdot (-10{,}140)}}{2 \cdot 3} \\ n = \frac{11 \pm \sqrt{121 + 121{,}680}}{6} \\ n = \frac{11 \pm \sqrt{121{,}801}}{6} \\ n = \frac{11 \pm 349}{6} \] 2. Calculate the roots: \[ n_1 = \frac{11 + 349}{6} = \frac{360}{6} = 60 \\ n_2 = \frac{11 - 349}{6} = \frac{-338}{6} \approx -56.33 \quad (\text{Not relevant since } n > 0) \] 3. Determine the intervals where the inequality holds: Since the quadratic coefficient is positive, the inequality \( 3n^2 - 11n - 10{,}140 > 0 \) holds for: \[ n < -56.33 \quad \text{or} \quad n > 60 \] Only \( n > 60 \) is relevant. 4. Since \( n \) must be an integer, the smallest integer greater than 60 is 61. **Answer:** The smallest value of \( n \) is **61**. ---