Maihematics' Iivestigation NSC - Grade 12 NW/ Feb 2025 A (2) 1.1.2 Make a coniecture with regard to \( r \) " and \( S_{n} \) as \( n \rightarrow \infty \) (2) 1.4.1 What is the sum of the geometric series \[ S_{-}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=1 \text { ? } \] 1.4.2 Make a conjecture with regard to \( r^{n} \) and \( S_{n} \) as \( n \rightarrow \infty \) (3) \[ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=-1 ? \] 1.5.2 Do the sums above approach some finite particular number as \( n \rightarrow \infty \) (1) i.e. is the sequence divergent or convergent? 1.6 CASE 5: \( -1

Ask by Stanley Davey. in South Africa

Feb 26,2025

Let's break down the problem step by step, focusing on the mathematical concepts involved. ### 1.1.2 Conjecture about \( r \) and \( S_n \) as \( n \rightarrow \infty \) In a geometric series, the sum \( S_n \) is given by: \[ S_n = a + ar + ar^2 + \ldots + ar^{n-1} \] The formula for the sum of the first \( n \) terms of a geometric series is: \[ S_n = a \frac{1 - r^n}{1 - r} \quad \text{if } r \neq 1 \] As \( n \rightarrow \infty \): - If \( |r| < 1 \), then \( r^n \rightarrow 0 \) and \( S_n \rightarrow \frac{a}{1 - r} \). - If \( |r| = 1 \), then \( S_n \) diverges (for \( r = 1 \), \( S_n \) becomes \( na \)). - If \( |r| > 1 \), then \( S_n \) diverges as well. **Conjecture**: As \( n \rightarrow \infty \), if \( -1 < r < 1 \), then \( S_n \) converges to \( \frac{a}{1 - r} \). ### 1.4.1 Sum of the geometric series when \( r = 1 \) If \( r = 1 \), the series becomes: \[ S_n = a + a + a + \ldots + a = na \] Thus, the sum diverges as \( n \rightarrow \infty \). ### 1.4.2 Conjecture about \( r^n \) and \( S_n \) as \( n \rightarrow \infty \) For \( r = -1 \): \[ S_n = a + (-a) + a + (-a) + \ldots \] This series oscillates between \( a \) and \( 0 \) depending on whether \( n \) is even or odd. Therefore, it does not converge to a single value. **Conjecture**: As \( n \rightarrow \infty \), \( S_n \) does not converge for \( r = -1 \). ### 1.5.2 Convergence of the sums For \( -1 < r < 1 \): - The series converges to \( \frac{a}{1 - r} \) as \( n \rightarrow \infty \). - For \( r = 1 \) and \( r = -1 \), the series diverges. ### 1.6.1 Possible values of \( r \) such that \( -1 < r < 1 \) Three possible values of \( r \) are: 1. \( r = 0.5 \) 2. \( r = -0.5 \) 3. \( r = 0.9 \) ### 1.6.2 to 1.6.6 Steps with the paper 1. **Step 1**: Cut the A4 paper into two equal rectangles, each with an area of \( 8 \, \text{unit}^2 \). 2. **Step 2**: Place one rectangle on the desktop and cut the other into two equal rectangles, each with an area of \( 4 \, \text{unit}^2 \). 3. **Step 3**: Place one of the \( 4 \, \text{unit}^2 \) rectangles on the desktop and cut the other into two equal rectangles, each with an area of \( 2 \, \text{unit}^2 \). 4. **Step 4**: Continue this process, cutting each remaining rectangle into two equal parts until it becomes too small to cut. 5. **Step 5**: The first piece of paper placed on the desktop has an area of \( 8 \, \text{unit}^2 \). This process illustrates the concept of halving areas, which can be related to the convergence of geometric series as the area approaches zero. ### Conclusion The investigation covers the behavior of geometric series under different conditions of \( r \) and explores the concept of convergence through practical steps involving cutting paper.