7 Determine the number of terms: a $$ \quad x(1+i) ; x(1+i)^{2} ; x(1+i)^{3} ; \ldots ; x(1+i)^{40} $$ b $$ \quad x(1+i)^{-1} ; x(1+i)^{-2} ; x(1+i)^{-3} ; \ldots ; x(1+i)^{-160} $$ c $$ an x ; \sin x ; \sin x \cos x ; \ldots ; an x(\cos x)^{9} $$ d $$ 1 ; \sin x ; 1-\cos ^{2} x ; \ldots ;(\sin x)^{11} $$ 8 Given $$ \mathrm{T}_{n}=\frac{3}{8}(r)^{n-1} $$ for a geometric sequence. a If the 4th term is $$ \frac{81}{8} $$, calculate the value of $$ r $$. b Which term is equal to $$ \frac{59049}{8} $$ ?

Question

7 Determine the number of terms: a $$ \quad x(1+i) ; x(1+i)^{2} ; x(1+i)^{3} ; \ldots ; x(1+i)^{40} $$ b $$ \quad x(1+i)^{-1} ; x(1+i)^{-2} ; x(1+i)^{-3} ; \ldots ; x(1+i)^{-160} $$ c $$ 	an x ; \sin x ; \sin x \cos x ; \ldots ; 	an x(\cos x)^{9} $$ d $$ 1 ; \sin x ; 1-\cos ^{2} x ; \ldots ;(\sin x)^{11} $$ 8 Given $$ \mathrm{T}_{n}=\frac{3}{8}(r)^{n-1} $$ for a geometric sequence. a If the 4th term is $$ \frac{81}{8} $$, calculate the value of $$ r $$. b Which term is equal to $$ \frac{59049}{8} $$ ?

UpStudy AI · Accepted Answer

Let's tackle each part of Problems **7** and **8** step by step.

---

## **Problem 7: Determine the Number of Terms**

### **a. Sequence:**
$$ x(1+i), \ x(1+i)^{2}, \ x(1+i)^{3}, \ \ldots, \ x(1+i)^{40} $$

- **Analysis:** The exponents start at 1 and go up to 40.
- **Number of Terms:** $$ 40 - 1 + 1 = 40 $$

**Answer:** **40 terms**

---

### **b. Sequence:**
$$ x(1+i)^{-1}, \ x(1+i)^{-2}, \ x(1+i)^{-3}, \ \ldots, \ x(1+i)^{-160} $$

- **Analysis:** The exponents start at $$-1$$ and decrement by 1 each time until $$-160$$.
- **Number of Terms:** $$ |-160 - (-1)| + 1 = 160 $$

**Answer:** **160 terms**

---

### **c. Sequence:**
$$ \tan x, \ \sin x, \ \sin x \cos x, \ \ldots, \ \tan x (\cos x)^9 $$

- **Analysis:** Observing the pattern:
  - First Term: $$ \tan x = \frac{\sin x}{\cos x} $$ (cos exponent = $$-1$$)
  - Second Term: $$ \sin x $$ (cos exponent = $$0$$)
  - Third Term: $$ \sin x \cos x $$ (cos exponent = $$1$$)
  - ...
  - Last Term: $$ \tan x (\cos x)^9 = \frac{\sin x}{\cos x} \cdot (\cos x)^9 = \sin x (\cos x)^8 $$ (cos exponent = $$9$$)
- **Exponents of $$\cos x$$:** From $$-1$$ to $$9$$, inclusive.
- **Number of Terms:** $$ 9 - (-1) + 1 = 11 $$

**Answer:** **11 terms**

---

### **d. Sequence:**
$$ 1, \ \sin x, \ 1 - \cos^2 x, \ \ldots, \ (\sin x)^{11} $$

- **Analysis:**
  - Recognize that $$ 1 - \cos^2 x = \sin^2 x $$
  - The sequence progresses as powers of $$\sin x$$ from $$0$$ to $$11$$.
- **Number of Terms:** $$ 11 - 0 + 1 = 12 $$

**Answer:** **12 terms**

---

## **Problem 8: Geometric Sequence Analysis**

Given the general term of a geometric sequence:
$$ \mathrm{T}_n = \frac{3}{8} r^{n-1} $$

### **a. If the 4th term is $$ \frac{81}{8} $$, calculate the value of $$ r $$.**

- **Given:** $$ \mathrm{T}_4 = \frac{3}{8} r^{4-1} = \frac{81}{8} $$
  
  $$
  \frac{3}{8} r^3 = \frac{81}{8}
  $$
  
- **Solving for $$ r^3 $$:**
  
  $$
  r^3 = \frac{81}{8} \div \frac{3}{8} = 27
  $$
  
- **Taking the cube root:**
  
  $$
  r = \sqrt[3]{27} = 3
  $$

**Answer:** **$$ r = 3 $$**

---

### **b. Which term is equal to $$ \frac{59049}{8} $$?**

- **Given:** $$ \mathrm{T}_n = \frac{3}{8} \times 3^{n-1} $$
  
  $$
  \frac{3}{8} \times 3^{n-1} = \frac{59049}{8}
  $$
  
- **Simplifying:**
  
  $$
  3^{n} = 59049
  $$
  
- **Recognizing powers of 3:**
  
  $$
  3^{10} = 59049
  $$
  
- **Therefore:**
  
  $$
  n = 10
  $$

**Answer:** **It is the 10th term**

---
**Problem 7:** - **a.** 40 terms - **b.** 160 terms - **c.** 11 terms - **d.** 12 terms **Problem 8:** - **a.** $$ r = 3 $$ - **b.** 10th term

7 Determine the number of terms:
a
b
c
d
8 Given for a geometric sequence.
a If the 4th term is , calculate the value of .
b Which term is equal to ?

Upstudy AI Solution

Answer

Solution

Problem 7: Determine the Number of Terms

a. Sequence:

b. Sequence:

c. Sequence:

d. Sequence:

Problem 8: Geometric Sequence Analysis

a. If the 4th term is , calculate the value of .

b. Which term is equal to ?

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7 Determine the number of terms: a b c d 8 Given for a geometric sequence. a If the 4th term is , calculate the value of . b Which term is equal to ?