Question 1 1.1. Debeanine the largest valuc of $$ k $$ for which: $$ \sum_{n=1}^{k} 2(3)^{n}

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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)

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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**. --- **Question 1:** 1. **Find the largest value of $$ k $$ for which:** $$ \sum_{n=1}^{k} 2 \cdot 3^{n} < 30{,}000 $$ **Answer:** $$ k = 8 $$ 2. **For which values of $$ x $$ will the series $$ \sum_{k=1}^{\infty} (4x - 1)^{k} $$ exist?** **Answer:** $$ 0 < x < \dfrac{1}{2} $$ **Question 2:** 1. **Consider the arithmetic sequence: $$ -8, -2, 4, 10, \ldots $$.** a. **If the $$ n^{\text{th}} $$ term is 148, find $$ n $$.** **Answer:** $$ n = 27 $$ b. **Find the smallest $$ n $$ where the sum of the first $$ n $$ terms is greater than 10,140.** **Answer:** $$ n = 61 $$

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