(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 $$ S_{k}1000 $$.

Question

(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 $$ S_{k}>1000 $$.

UpStudy AI · Accepted Answer

Let's analyze the given geometric sequence and solve each part step by step.

### Given Sequence:
$$ \frac{3}{4}, \frac{3}{2}, 3, \ldots $$

### Common Ratio ($$ r $$):
To find the common ratio $$ r $$, divide the second term by the first term:
$$
r = \frac{\frac{3}{2}}{\frac{3}{4}} = \frac{3}{2} \times \frac{4}{3} = 2
$$

Thus, the sequence is:
$$
\frac{3}{4}, \frac{3}{2}, 3, 6, 12, 24, \ldots \quad \text{with } r = 2
$$

### (1) Sum of the First 10 Terms ($$ S_{10} $$):
The sum of the first $$ n $$ terms of a geometric sequence is given by:
$$
S_n = a_1 \frac{r^n - 1}{r - 1}
$$
where $$ a_1 = \frac{3}{4} $$, $$ r = 2 $$, and $$ n = 10 $$.

Plugging in the values:
$$
S_{10} = \frac{3}{4} \times \frac{2^{10} - 1}{2 - 1} = \frac{3}{4} \times (1024 - 1) = \frac{3}{4} \times 1023 = \frac{3069}{4} = 767 \frac{1}{4}
$$

### (2) Number of Terms Adding Up to $$ \frac{12285}{4} $$ ($$ n $$):
Set the sum formula equal to $$ \frac{12285}{4} $$:
$$
\frac{3}{4} (2^n - 1) = \frac{12285}{4}
$$
Multiply both sides by 4:
$$
3(2^n - 1) = 12285
$$
Divide by 3:
$$
2^n - 1 = 4095
$$
Add 1:
$$
2^n = 4096
$$
Recognizing that $$ 4096 = 2^{12} $$, we find:
$$
n = 12
$$

### (3) Smallest $$ k $$ for Which $$ S_k > 1000 $$:
We need to find the smallest $$ k $$ such that:
$$
\frac{3}{4} (2^k - 1) > 1000
$$
Multiply both sides by $$ \frac{4}{3} $$:
$$
2^k - 1 > \frac{4000}{3} \approx 1333.33
$$
Add 1:
$$
2^k > 1334.33
$$
Determine the smallest $$ k $$ where $$ 2^k $$ exceeds 1334.33:
$$
2^{10} = 1024 \quad (\text{Too small}) \
2^{11} = 2048 \quad (\text{Sufficient})
$$
Thus, the smallest $$ k $$ is:
$$
k = 11
$$

### **Final Answers:**
1. The sum of the first 10 terms is $$ \frac{3069}{4} $$.
2. **12** terms add up to $$ \frac{12285}{4} $$.
3. The smallest value of $$ k $$ for which $$ S_k > 1000 $$ is **11**.
1. The sum of the first 10 terms is 767.25. 2. 12 terms add up to 3071.25. 3. The smallest value of $$ k $$ for which $$ S_k > 1000 $$ is 11.

(e) Consider the geometric sequence
(1) Calculate the sum of the first 10 terms of the sequence.
(2) How many terms in this sequence will add up to ?
(3) Determine the smallest value of for which .

Upstudy AI Solution

Answer

Solution

Given Sequence:

Common Ratio ():

(1) Sum of the First 10 Terms ():

(2) Number of Terms Adding Up to ():

(3) Smallest for Which :

Final Answers:

The Deep Dive

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(e) Consider the geometric sequence (1) Calculate the sum of the first 10 terms of the sequence. (2) How many terms in this sequence will add up to ? (3) Determine the smallest value of for which .