(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 \( S_{k}>1000 \).

Let's analyze the geometric sequence given: \[ \frac{3}{4}, \frac{3}{2}, 3, \ldots \] **Common Ratio (\( r \)):** To find the common ratio \( r \), divide the second term by the first term: \[ r = \frac{\frac{3}{2}}{\frac{3}{4}} = 2 \] So, the sequence is geometric with first term \( a = \frac{3}{4} \) and common ratio \( r = 2 \). ### (1) **Sum of the First 10 Terms** The sum of the first \( n \) terms of a geometric sequence is given by: \[ S_n = a \frac{r^n - 1}{r - 1} \] For \( n = 10 \): \[ S_{10} = \frac{3}{4} \cdot \frac{2^{10} - 1}{2 - 1} = \frac{3}{4} \cdot (1024 - 1) = \frac{3}{4} \cdot 1023 = \frac{3069}{4} = 767.25 \] **Answer:** The sum of the first 10 terms is \( \frac{3069}{4} \) or 767.25. ### (2) **Number of Terms Adding Up to \( \frac{12285}{4} \)** We need to find \( n \) such that: \[ S_n = \frac{12285}{4} \] Using the sum formula: \[ \frac{3}{4} \cdot (2^n - 1) = \frac{12285}{4} \] Multiply both sides by 4 to eliminate the denominator: \[ 3(2^n - 1) = 12285 \] Divide by 3: \[ 2^n - 1 = 4095 \quad \Rightarrow \quad 2^n = 4096 \] Since \( 2^{12} = 4096 \): **Answer:** 12 terms are needed. ### (3) **Smallest \( k \) for Which \( S_k > 1000 \)** We need to find the smallest \( k \) such that: \[ \frac{3}{4} \cdot (2^k - 1) > 1000 \] Multiply both sides by \( \frac{4}{3} \): \[ 2^k - 1 > \frac{4000}{3} \approx 1333.\overline{3} \] Thus: \[ 2^k > 1334.\overline{3} \] Finding the smallest integer \( k \): \[ 2^{10} = 1024 \quad \text{and} \quad 2^{11} = 2048 \] Since \( 2048 > 1334.\overline{3} \): **Answer:** The smallest value of \( k \) is 11. ### **Summary of Answers:** 1. **Sum of the first 10 terms:** \( \frac{3069}{4} \) or 767.25. 2. **Number of terms adding up to \( \frac{12285}{4} \):** 12 terms. 3. **Smallest \( k \) with \( S_k > 1000 \):** \( k = 11 \).