QUESTION 1 1.1 Given: \( \mathrm{S}_{n}=\frac{a\left(r^{n}-1\right)}{r-1} \) 1.1 .1 Write \( S_{n} \) in expanded form 1.1 .2 Which term in 1.1 .1 is affected by the Consider \( r^{n} \) (write answer in scientific notation f 1.2 CASE \( 1: r>1 \) 1.2 .1 If \( r=1,001 \) determine the values of (a) \( r^{200} \) and \( \mathrm{S}_{200} \) (b) \( r^{20000} \) and \( \mathrm{S}_{20000} \) 1.2 .2 Determine \( r^{200} \) and \( \mathrm{S}_{200} \) if: (a) \( r=\frac{5}{2} \) (b) \( r=3 \)\( ~ ل ~\left(\begin{array}{l}\text { (b) }\end{array}\right. \)

Let's solve the problem step by step. ### 1.1 Given: \[ \mathrm{S}_{n}=\frac{a\left(r^{n}-1\right)}{r-1} \] #### 1.1.1 Write \( S_{n} \) in expanded form To expand \( S_n \), we can distribute the terms in the formula. The expression can be rewritten as: \[ S_n = \frac{a \cdot r^n - a}{r - 1} \] This shows that \( S_n \) consists of two main components: \( a \cdot r^n \) and \( -a \). #### 1.1.2 Which term in 1.1.1 is affected by \( r^{n} \)? The term that is affected by \( r^{n} \) is \( a \cdot r^n \). This term grows exponentially as \( n \) increases, especially when \( r > 1 \). ### 1.2 CASE \( 1: r > 1 \) #### 1.2.1 If \( r = 1.001 \), determine the values of: ##### (a) \( r^{200} \) and \( S_{200} \) First, we calculate \( r^{200} \): \[ r^{200} = (1.001)^{200} \] Next, we calculate \( S_{200} \): \[ S_{200} = \frac{a \left( (1.001)^{200} - 1 \right)}{1.001 - 1} \] ##### (b) \( r^{20000} \) and \( S_{20000} \) Now, we calculate \( r^{20000} \): \[ r^{20000} = (1.001)^{20000} \] Next, we calculate \( S_{20000} \): \[ S_{20000} = \frac{a \left( (1.001)^{20000} - 1 \right)}{1.001 - 1} \] #### 1.2.2 Determine \( r^{200} \) and \( S_{200} \) if: ##### (a) \( r = \frac{5}{2} \) First, we calculate \( r^{200} \): \[ r^{200} = \left( \frac{5}{2} \right)^{200} \] Next, we calculate \( S_{200} \): \[ S_{200} = \frac{a \left( \left( \frac{5}{2} \right)^{200} - 1 \right)}{\frac{5}{2} - 1} \] ##### (b) \( r = 3 \) First, we calculate \( r^{200} \): \[ r^{200} = 3^{200} \] Next, we calculate \( S_{200} \): \[ S_{200} = \frac{a \left( 3^{200} - 1 \right)}{3 - 1} \] Now, let's perform the calculations for \( r^{200} \), \( S_{200} \), \( r^{20000} \), and \( S_{20000} \) using the specified values of \( r \). I'll calculate these values now. Simplify the expression by following steps: - step0: Evaluate the power: \(1.001^{20000}\) - step1: Convert the expressions: \(\left(\frac{1001}{1000}\right)^{20000}\) - step2: Evaluate the power: \(\frac{1001^{20000}}{1000^{20000}}\) Expand the expression \( (1.001)^{200} \) Simplify the expression by following steps: - step0: Evaluate the power: \(1.001^{200}\) - step1: Convert the expressions: \(\left(\frac{1001}{1000}\right)^{200}\) - step2: Evaluate the power: \(\frac{1001^{200}}{1000^{200}}\) Expand the expression \( 3^{200} \) Simplify the expression by following steps: - step0: : \(3^{200}\) Expand the expression \( (\frac{5}{2})^{200} \) Simplify the expression by following steps: - step0: Evaluate the power: \(\left(\frac{5}{2}\right)^{200}\) - step1: Evaluate the power: \(\frac{5^{200}}{2^{200}}\) Here are the calculated values for the specified cases: ### 1.2.1 If \( r = 1.001 \): ##### (a) \( r^{200} \) and \( S_{200} \) - \( r^{200} = (1.001)^{200} \approx 1.221386 \) (in scientific notation: \( \approx 1.221 \times 10^{0} \)) - To calculate \( S_{200} \), we need to substitute \( r^{200} \) into the formula: \[ S_{200} = \frac{a \left( (1.001)^{200} - 1 \right)}{1.001 - 1} = \frac{a \left( 1.221386 - 1 \right)}{0.001} = \frac{a \cdot 0.221386}{0.001} \approx 221.386a \] ##### (b) \( r^{20000} \) and \( S_{20000} \) - \( r^{20000} = (1.001)^{20000} \approx 4.803409 \times 10^{8} \) - To calculate \( S_{20000} \): \[ S_{20000} = \frac{a \left( (1.001)^{20000} - 1 \right)}{1.001 - 1} = \frac{a \left( 4.803409 \times 10^{8} - 1 \right)}{0.001} \approx 4.803409 \times 10^{11} a \] ### 1.2.2 Determine \( r^{200} \) and \( S_{200} \) if: ##### (a) \( r = \frac{5}{2} \) - \( r^{200} = \left( \frac{5}{2} \right)^{200} \approx 3.872592 \times 10^{79} \) - To calculate \( S_{200} \): \[ S_{200} = \frac{a \left( \left( \frac{5}{2} \right)^{200} - 1 \right)}{\frac{5}{2} - 1} = \frac{a \left( 3.872592 \times 10^{79} - 1 \right)}{\frac{3}{2}} \approx \frac{2a \cdot (3.872592 \times 10^{79} - 1)}{3} \approx \frac{2a \cdot 3.872592 \times 10^{79}}{3} \approx 2.582 \times 10^{79} a \] ##### (b) \( r = 3 \) - \( r^{200} = 3^{200} \approx 2.65614 \times 10^{95} \) - To calculate \( S_{200} \): \[ S_{200} = \frac{a \left( 3^{200} - 1 \right)}{3 - 1} = \frac{a \left( 2.65614 \times 10^{95} - 1 \right)}{2} \approx \frac{a \cdot 2.65614 \times 10^{95}}{2} \approx 1.32807 \times 10^{95} a \] ### Summary of Results: - For \( r = 1.001 \): - \( r^{200} \approx 1.221 \times 10^{0} \) - \( S_{200} \approx 221.386a \) - \( r^{20000} \approx 4.803409 \times 10^{8} \) - \( S_{20000} \approx 4.803409 \times 10^{11} a \) - For \( r = \frac{5}{2} \): - \( r^{200} \approx 3.872592 \times 10^{79} \) - \( S_{200} \approx 2.582 \times 10^{79} a \) - For \( r = 3 \): - \( r^{200} \approx 2.65614 \times 10^{95} \) - \( S_{200} \approx 1.32807 \times 10^{95} a \)