(3) How many terms are there insider the geometric sequence .
(1) Calculate .
(2) Determine the value of if .

Ask by Peters Tucker. in South Africa

Jan 22,2025

Question

(3) How many terms are there insider the geometric sequence $$ -320 ; 160 ;-80 ; \ldots ;-\frac{5}{64} $$. (1) Calculate $$ T_{8} $$. (2) Determine the value of $$ k $$ if $$ T_{k}=-\frac{5}{16} $$.

UpStudy AI · Accepted Answer

Let's analyze the given geometric sequence: $$-320, 160, -80, \ldots, -\frac{5}{64}$$.

### Common Ratio ($$r$$):
To find the common ratio $$ r $$, divide the second term by the first term:
$$
r = \frac{160}{-320} = -0.5
$$

### General Term ($$T_n$$):
The $$ n $$-th term of a geometric sequence is given by:
$$
T_n = T_1 \cdot r^{n-1} = -320 \cdot (-0.5)^{n-1}
$$

---

### (3) Number of Terms in the Sequence:
We need to find $$ n $$ such that:
$$
T_n = -\frac{5}{64}
$$
Substitute into the general term:
$$
-320 \cdot (-0.5)^{n-1} = -\frac{5}{64}
$$
Divide both sides by $$-320$$:
$$
(-0.5)^{n-1} = \frac{-5}{64} \div (-320) = \frac{5}{20480} = \frac{1}{4096}
$$
Recognize that $$ \frac{1}{4096} = \left(\frac{1}{2}\right)^{12} $$, so:
$$
(-0.5)^{n-1} = \left(-\frac{1}{2}\right)^{n-1} = \left(\frac{1}{2}\right)^{12} \quad \text{and} \quad (-1)^{n-1} = 1
$$
This implies:
$$
n - 1 = 12 \quad \Rightarrow \quad n = 13
$$
**Answer:** There are **13 terms** in the sequence.

---

### (1) Calculating $$ T_8 $$:
Using the general term:
$$
T_8 = -320 \cdot (-0.5)^{7} = -320 \cdot \left(-\frac{1}{128}\right) = \frac{320}{128} = 2.5 = \frac{5}{2}
$$
**Answer:** $$ T_8 = \frac{5}{2} $$

---

### (2) Determining $$ k $$ for $$ T_k = -\frac{5}{16} $$:
Set the general term equal to $$-\frac{5}{16}$$:
$$
-320 \cdot (-0.5)^{k-1} = -\frac{5}{16}
$$
Divide both sides by $$-320$$:
$$
(-0.5)^{k-1} = \frac{-5}{16} \div (-320) = \frac{5}{5120} = \frac{1}{1024} = \left(\frac{1}{2}\right)^{10}
$$
Thus:
$$
(k - 1) = 10 \quad \Rightarrow \quad k = 11
$$
**Answer:** The value of $$ k $$ is **11**.

---

### **Summary of Answers:**
1. **Number of terms:** 13
2. **$$ T_8 = \frac{5}{2} $$**
3. **$$ k = 11 $$**
There are 13 terms in the sequence. $$ T_8 = \frac{5}{2} $$. When $$ T_k = -\frac{5}{16} $$, $$ k = 11 $$.

(3) How many terms are there insider the geometric sequence .
(1) Calculate .
(2) Determine the value of if .

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