1.3.2 Make a conjecture with regard to $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$ (2) 1.4 CASE 3: $$ r=1 $$ 1.4.1 What is the sum of the geometric series $$ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=1 \text { ? } $$ 1.4.2 Make a conjecture with regard to $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$ (2) 1.5 CASE 4: $$ r=-1 $$ 1.5.1 What is the sum of the geometric series $$ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=-1 ? $$ 1.5.2 Do the sums above approach some finite particular number as $$ n \rightarrow \infty $$ i.e. is the sequence divergent or convergent? 1.6 CASE 5: $$ -1

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Below is one sample solution that addresses each part of the question. You may write your answers on an A4 paper using the “provided grid” and following the cutting/folding instructions. Notice that some parts ask for a written “conjecture” – your answer may be phrased in words with a brief explanation. ────────────────────────────── 1.3.2 Conjecture (Geometric Series for arbitrary r) • Conjecture for rⁿ as n → ∞: – If |r| < 1, then rⁿ becomes smaller and approaches 0. – If |r| > 1, then rⁿ grows without bound (tends to infinity or minus infinity depending on the sign). – If r = 1, then rⁿ always equals 1. – If r = –1, then rⁿ oscillates between 1 and –1. • Conjecture for Sₙ, the sum of the first n terms: – When |r| < 1, the finite sum Sₙ = a(1 – rⁿ)/(1 – r) approaches a/(1 – r) as n → ∞ (because rⁿ → 0). – When |r| > 1 or r = –1, the sum does not settle toward a fixed finite number, so we say the series is divergent (except in the trivial case a = 0). – When r = 1 the sum Sₙ = a + a + … + a = n · a, so if a ≠ 0 it grows without bound. ────────────────────────────── 1.4 CASE 3: r = 1 1.4.1 Sum of the series when r = 1 The series is Sₙ = a + a·1 + a·1² + ⋯ + a·1^(n–1). Since 1 raised to any power is 1, we have: Sₙ = a + a + a + ⋯ + a (n terms), so Sₙ = n · a. 1.4.2 Conjecture for r = 1 as n → ∞ Since r = 1, rⁿ = 1 for every n. Also, Sₙ = n · a: – If a ≠ 0, then as n increases, n · a becomes arbitrarily large (or arbitrarily negative if a < 0). – Thus, the sum does not approach a finite number but diverges. – In summary, when r = 1 with a ≠ 0, the series diverges; the n‑th term stays constant while the sum increases without bound. ────────────────────────────── 1.5 CASE 4: r = –1 1.5.1 Sum of the series when r = –1 The series is Sₙ = a + a(–1) + a(–1)² + ⋯ + a(–1)^(n–1). Notice the pattern: – For n = 1, S₁ = a. – For n = 2, S₂ = a + (–a) = 0. – For n = 3, S₃ = a – a + a = a. – For n = 4, S₄ = a – a + a – a = 0. Thus, the sum alternates between a (when n is odd) and 0 (when n is even). 1.5.2 Convergence or divergence for r = –1? Because Sₙ oscillates between two values (a and 0) and does not approach one single number as n → ∞, the sequence of partial sums does not converge. Therefore, the geometric series (for nonzero a) is divergent in this case. ────────────────────────────── 1.6 CASE 5: –1 < r < 1 1.6.1 Write THREE possible values of r Examples of r in the interval (–1, 1) include: • r = 0.5 • r = –0.75 • r = 0.2 1.6.2 Step 1 (Paper cut): • Begin with one A4 paper. • Cut the A4 paper along its longest side into two equal rectangles. • Each resulting rectangle is defined to have an area of 16 unit². (Thus, the whole A4 paper has an area of 32 unit².) 1.6.3 Step 2 (Paper cut): • Place one of the two 16 unit² rectangles on the desktop. • Take the other 16 unit² rectangle and cut it along its longest side into two equal rectangles. • Each of these two new rectangles has an area of 16/2 = 8 unit². 1.6.4 Step 3 (Paper cut): • Now take one of the 8 unit² rectangles and place it on the desktop. • Cut the remaining 8 unit² rectangle along its longest side into two equal parts. • Each of these pieces has an area of 8/2 = 4 unit². 1.6.5 Step 4 (Continue the process): • Continue the procedure: at each step, set aside one piece and cut the last piece you hold (which always has been the “other half”) into two equal rectangles. • Repeat this until the paper becomes too small to continue the folding/cutting process. 1.6.6 Step 5 (List the areas): The procedure shows that: – The first piece placed on the desktop has an area equal to ½ of the A4 paper, that is, 16 unit². – The second piece placed (from the first cut of the other half) has an area of ¼ of the A4 paper, i.e. 8 unit². – The next three pieces will have areas that are half of the previous piece each time. Thus: • Third piece: 1/8 of the A4, which is 16/4 = 4 unit². • Fourth piece: 1/16 of the A4, which is 16/8 = 2 unit². • Fifth piece: 1/32 of the A4, which is 16/16 = 1 unit². (Alternatively, you can express these as fractions of the A4 area: 1/2, 1/4, 1/8, 1/16, 1/32.) 1.6.7 Explanation: Why these areas form a geometric sequence Each time you cut the piece of paper, the area of the piece you produce is exactly one-half the area of the piece that was just cut. In other words, if you denote the area of the first piece by A (with A = 1/2 of A4), each subsequent area is obtained by multiplying by the constant ratio r = 1/2. Therefore, the areas form a geometric sequence where: A, A·(1/2), A·(1/2)², A·(1/2)³, etc. Because the ratio between successive terms is equal throughout (namely, 1/2), this is the definition of a geometric sequence. ────────────────────────────── Additional Notes • In summary, these examples illustrate the behavior of a geometric series for different choices of r: – |r| < 1 leads to convergence of the series to a/(1–r), and the individual term rⁿ → 0. – r = 1 gives a constantly repeated term (thus a linear sum in n). – r = –1 produces oscillation. – |r| > 1 (though not shown in the paper-cut activity) results in divergence. You may now record your concise answers on the provided A4 grid. **1.3.2 Conjecture:** - If $$ |r| < 1 $$, then $$ r^{n} $$ approaches 0 as $$ n $$ increases. - If $$ |r| > 1 $$, then $$ r^{n} $$ grows without bound. - If $$ r = 1 $$, $$ r^{n} $$ remains 1. - If $$ r = -1 $$, $$ r^{n} $$ alternates between 1 and -1. **1.4 CASE 3: $$ r = 1 $$** - The sum $$ S_{n} = n imes a $$. - As $$ n $$ increases, $$ S_{n} $$ grows without bound if $$ a eq 0 $$. **1.5 CASE 4: $$ r = -1 $$** - The sum alternates between $$ a $$ and 0. - The series does not approach a finite number and is divergent. **1.6 CASE 5: $$ -1 < r < 1 $$** - Three possible values of $$ r $$: 0.5, -0.75, 0.2. - Cutting the A4 paper in half each time creates areas that form a geometric sequence: 16, 8, 4, 2, 1 unit². **1.6.7 Explanation:** Each subsequent piece of paper has half the area of the previous one, forming a geometric sequence with a common ratio of 1/2.

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