(e) Consider the geometric sequence \( \frac{3}{4} ; \frac{3}{2} ; 3: \ldots \) (1) Calculate the sum of the first 10 terms of the sequence. (2) How many terms in this sequence will add up to \( \frac{12285}{4} \) ? (3) Determine the smallest value of \( k \) for which \( S_{k}>1000 \).

To solve the problem, let's break it down step by step! **(1) Calculate the sum of the first 10 terms of the sequence.** The given sequence is a geometric sequence where the first term \( a = \frac{3}{4} \) and the common ratio \( r = \frac{\frac{3}{2}}{\frac{3}{4}} = 2 \). The sum of the first \( n \) terms of a geometric series is given by the formula: \[ S_n = a \frac{(1 - r^n)}{1 - r} \] Substituting for \( n = 10 \): \[ S_{10} = \frac{3}{4} \cdot \frac{1 - 2^{10}}{1 - 2} = \frac{3}{4} \cdot \frac{1 - 1024}{-1} = \frac{3}{4} \cdot 1023 = \frac{3069}{4} \] Thus, the sum of the first 10 terms is \( \frac{3069}{4} \). --- **(2) How many terms in this sequence will add up to \( \frac{12285}{4} \)?** Using the sum formula \( S_n \) again, equate it to \( \frac{12285}{4} \): \[ \frac{3}{4} \cdot \frac{1 - 2^n}{1 - 2} = \frac{12285}{4} \] This simplifies to: \[ \frac{3}{4} \cdot (1 - 2^n) = -\frac{12285}{4} \] Multiplying both sides by 4: \[ 3(1 - 2^n) = -12285 \] \[ 3 - 3 \cdot 2^n = -12285 \implies 3 \cdot 2^n = 3 + 12285 = 12288 \] \[ 2^n = \frac{12288}{3} = 4096 \implies 2^n = 2^{12} \] Thus, \( n = 12 \). Therefore, 12 terms in this sequence will add up to \( \frac{12285}{4} \). --- **(3) Determine the smallest value of \( k \) for which \( S_{k}>1000 \).** Using the sum formula again, we want to find \( k \) such that: \[ S_k > 1000 \] Substituting into the formula: \[ \frac{3}{4} \cdot \frac{1 - 2^k}{1 - 2} > 1000 \] This simplifies to: \[ \frac{3}{4} (1 - 2^k) > -1000 \] Multiplying by \( -4 \) (remembering to flip the inequality): \[ 3(1 - 2^k) < 4000 \] \[ 3 - 3 \cdot 2^k < 4000 \implies -3 \cdot 2^k < 3997 \implies 3 \cdot 2^k > -3997 \] Since \( 2^k \) is always positive, we only need to solve for: \[ 2^k > \frac{3997}{3} \approx 1332.33 \] Taking the logarithm base 2 of both sides, we find: \[ k > \log_2(1332.33) \approx 10.4 \] Thus the smallest integer \( k = 11 \). So the smallest value of \( k \) such that \( S_k > 1000 \) is \( 11 \).