Sigma Notation

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Answer Key & Explanations

Sigma Notation · Grade 12 · Worksheet 1

1. A geometric pattern is formed by stacking circles in rows. The top row has 1 circle, the second row has 3 circles, the third row has 5 circles, and this pattern continues for 15 rows. Using sigma notation, write the series representing the total number of circles in this triangular arrangement, then evaluate the sum. Answer: 225 Solution: - Row 1: 1 circle - Row 2: 3 circles - Row 3: 5 circles - Pattern continues for 15 rows. We see that the number of circles in row \( n \) is \( 2n - 1 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the pattern** The problem says: - Row 1: 1 circle - Row 2: 3 circles - Row 3: 5 circles - Pattern continues for 15 rows. We see that the number of circles in row \( n \) is \( 2n - 1 \). Check: Row 1: \( 2(1) - 1 = 1 \) Row 2: \( 2(2) - 1 = 3 \) Row 3: \( 2(3) - 1 = 5 \) Yes, correct. --- **Step 2: Write the series in sigma notation** The total number of circles in 15 rows is: \[ \sum_{n=1}^{15} (2n - 1) \] --- **Step 3: Break the sum into two parts** \[ \sum_{n=1}^{15} (2n - 1) = 2 \sum_{n=1}^{15} n - \sum_{n=1}^{15} 1 \] --- **Step 4: Evaluate each sum** First, \( \sum_{n=1}^{15} n = \frac{15 \times 16}{2} = 120 \) So \( 2 \sum_{n=1}^{15} n = 2 \times 120 = 240 \) Second, \( \sum_{n=1}^{15} 1 = 15 \) --- **Step 5: Subtract** \[ 240 - 15 = 225 \] --- **Step 6: Conclusion** The total number of circles is \( 225 \). --- **Final Answer:** 225

2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = 50te^(-0.2t) milligrams per liter, where t is time in hours. The company needs to calculate the total drug exposure over the first 10 hours, which is represented by the definite integral of C(t) from 0 to 10. Express this total drug exposure using sigma notation with 5 subintervals of equal width. Answer: ∑_{i=1}^5 [2 * 50(2i-1)e^(-0.2(2i-1))] Solution: C(t) = 50 t e^{-0.2 t} We want the total drug exposure over the first 10 hours, which is: \int_{0}^{10} C(t) \, dt But instead of computing the integral exactly, we must express it in sigma notation with 5 subintervals of equal width — meaning a Riemann sum.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understanding the problem** We are given: \[ C(t) = 50 t e^{-0.2 t} \] We want the total drug exposure over the first 10 hours, which is: \[ \int_{0}^{10} C(t) \, dt \] But instead of computing the integral exactly, we must express it in **sigma notation with 5 subintervals of equal width** — meaning a Riemann sum. --- **Step 2: Setting up the Riemann sum** Number of subintervals: \( n = 5 \) Interval: \([0, 10]\) Width of each subinterval: \[ \Delta t = \frac{10 - 0}{5} = 2 \] So: \[ t_0 = 0, \quad t_1 = 2, \quad t_2 = 4, \quad t_3 = 6, \quad t_4 = 8, \quad t_5 = 10 \] --- **Step 3: Choosing sample points** The problem’s given answer uses the **midpoint rule** (since it has \(2i - 1\) inside, which gives midpoints when multiplied by \(\Delta t / 2\) appropriately — let's check). Midpoint of i-th subinterval: \[ t_i^* = \frac{t_{i-1} + t_i}{2} \] \[ t_{i-1} = 2(i-1), \quad t_i = 2i \] \[ t_i^* = \frac{2(i-1) + 2i}{2} = \frac{4i - 2}{2} = 2i - 1 \] So indeed, \( t_i^* = 2i - 1 \). --- **Step 4: Midpoint Riemann sum formula** Midpoint rule: \[ \int_{0}^{10} C(t) \, dt \approx \sum_{i=1}^{n} C(t_i^*) \cdot \Delta t \] Here \( \Delta t = 2 \), \( t_i^* = 2i - 1 \), \( C(t) = 50 t e^{-0.2 t} \). So: \[ C(t_i^*) = 50 (2i - 1) e^{-0.2 (2i - 1)} \] Multiply by \( \Delta t = 2 \): \[ \text{Term in sum} = 2 \cdot 50 (2i - 1) e^{-0.2 (2i - 1)} \] --- **Step 5: Final sigma notation** \[ \int_{0}^{10} C(t) \, dt \approx \sum_{i=1}^{5} \left[ 2 \cdot 50 (2i - 1) e^{-0.2 (2i - 1)} \right] \] This matches the given correct answer. --- **Final Answer:** ∑_{i=1}^5 [2 * 50(2i-1)e^(-0.2(2i-1))]

3. Mere is constructing a visual pattern using hexagonal tiles. The first layer has 2 hexagons. The second layer has 4 hexagons. The third layer has 6 hexagons, and this pattern continues for 12 layers. Using sigma notation, write the series representing the total number of hexagons in this arrangement, then evaluate the sum. Answer: 156 Solution: Identify the pattern. Layer 1 has 2 hexagons, layer 2 has 4 hexagons, layer 3 has 6 hexagons. The number of hexagons in layer n is 2n.
   Full step-by-step solution

   Step 1: Identify the pattern. Layer 1 has 2 hexagons, layer 2 has 4 hexagons, layer 3 has 6 hexagons. The number of hexagons in layer n is 2n. Check: n=1 gives 2, n=2 gives 4, n=3 gives 6. This is correct. Step 2: Write the total number of hexagons using sigma notation. For 12 layers, the sum is from n=1 to 12 of 2n. So S = sum from n=1 to 12 of 2n. Step 3: Factor out the constant 2. S = 2 * sum from n=1 to 12 of n. Step 4: Use the formula for the sum of the first n positive integers: sum from n=1 to 12 of n = 12 * 13 / 2 = 156 / 2 = 78. Step 5: Multiply by 2: S = 2 * 78 = 156. Step 6: The total number of hexagons is 156.

4. Noah is constructing a visual pattern using hexagonal tiles. The first layer (center) has 1 tile. The second layer surrounding it has 6 tiles. The third layer has 12 tiles, and the fourth layer has 18 tiles. This pattern continues such that each layer after the first increases by 6 tiles compared to the previous layer. Write the total number of tiles in the first 11 layers using sigma notation, then evaluate the sum. Answer: 331 Solution: Identify the pattern. Layer 1: 1 tile. Layer 2: 6 tiles.
   Full step-by-step solution

   Step 1: Identify the pattern. Layer 1: 1 tile. Layer 2: 6 tiles. Layer 3: 12 tiles. Layer 4: 18 tiles. Starting from layer 2, each layer adds 6 tiles: 6, 12, 18, ... This is an arithmetic sequence with first term 6 and common difference 6. Step 2: For layer n (where n >= 2), the number of tiles is 6(n-1). Check: n=2 gives 6(1)=6, n=3 gives 6(2)=12, correct. Step 3: Total tiles in 11 layers = layer 1 + sum from n=2 to 11 of 6(n-1). Let k = n-1, so when n=2, k=1; when n=11, k=10. Then sum from k=1 to 10 of 6k. Step 4: So total = 1 + sum from k=1 to 10 of 6k = 1 + 6 * sum from k=1 to 10 of k. Step 5: Sum from k=1 to 10 of k = 10*11/2 = 55. Step 6: 6 * 55 = 330. Then total = 1 + 330 = 331. Step 7: In sigma notation: 1 + sum_{k=1}^{10} 6k, or sum_{n=1}^{11} a_n where a_1=1 and a_n=6(n-1) for n>=2. The total number of tiles is 331.

5. Hana is a materials engineer testing the tensile strength of a new alloy. She measures the stress (in megapascals) required to elongate a sample by different amounts. The stress function is modeled by S(x) = 12x^2 - 5x + 8, where x is the strain (as a percentage) for 0 ≤ x ≤ 12. To calculate the total work done (in megapascal-percent) during the first 12% of strain, she needs to find the area under the stress-strain curve. Express this total work using sigma notation with 6 subintervals of equal width, using right endpoints for the Riemann sum approximation. Answer: ∑[i=1 to 6] (12(2i)^2 - 5(2i) + 8) × 2 Solution: The interval is from x=0 to x=12. Total width = 12 - 0 = 12. With 6 subintervals, each subinterval has width Δx = 12/6 = 2.
   Full step-by-step solution

   Step 1: The interval is from x=0 to x=12. Total width = 12 - 0 = 12. Step 2: With 6 subintervals, each subinterval has width Δx = 12/6 = 2. Step 3: Using right endpoints, the x-values are: x1 = 2, x2 = 4, x3 = 6, x4 = 8, x5 = 10, x6 = 12. Step 4: In general, for right endpoints: x_i = 0 + i × Δx = 2i, where i goes from 1 to 6. Step 5: The Riemann sum using sigma notation is: ∑[i=1 to 6] S(x_i) × Δx. Step 6: Substitute S(x) = 12x^2 - 5x + 8 and x_i = 2i, Δx = 2: ∑[i=1 to 6] (12(2i)^2 - 5(2i) + 8) × 2. Step 7: The final sigma notation expression is ∑[i=1 to 6] (12(2i)^2 - 5(2i) + 8) × 2. The answer is ∑[i=1 to 6] (12(2i)^2 - 5(2i) + 8) × 2.

6. Σ(k=1 to 9) (3k - 4) = ? Answer: 99 Solution: Identify the number of terms: n = 9. Find the first term (k=1): 3(1) - 4 = 3 - 4 = -1. Find the last term (k=9): 3(9) - 4 = 27 - 4 = 23.
   Full step-by-step solution

   Step 1: Identify the number of terms: n = 9. Step 2: Find the first term (k=1): 3(1) - 4 = 3 - 4 = -1. Step 3: Find the last term (k=9): 3(9) - 4 = 27 - 4 = 23. Step 4: Use the sum formula for an arithmetic series: S = n/2 * (first term + last term). Step 5: S = 9/2 * (-1 + 23) = 9/2 * 22 = 9 * 11 = 99. The answer is 99.