Sigma Notation

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Answer Key & Explanations

Sigma Notation · Grade 12 · Worksheet 1

1. Sophia is constructing a visual pattern using square tiles arranged in concentric square rings around a central tile. The central tile (ring 0) is a single tile. The first ring around it consists of 8 tiles forming a larger square. The second ring has 16 tiles, the third ring has 24 tiles, and this pattern continues such that each subsequent ring adds 8 more tiles than the previous ring. If the pattern has a total of 11 rings (ring 0 through ring 10), write the total number of tiles using sigma notation, then evaluate the sum. Answer: 441 Solution: Identify the pattern. Ring 0 has 1 tile. Ring 1 has 8 tiles.
   Full step-by-step solution

   Step 1: Identify the pattern. Ring 0 has 1 tile. Ring 1 has 8 tiles. Ring 2 has 16 tiles. Ring 3 has 24 tiles. For ring k where k >= 1, the number of tiles is 8k. So ring 1: 8(1)=8, ring 2: 8(2)=16, ring 3: 8(3)=24, etc. Step 2: Write the total number of tiles using sigma notation for 11 rings (k = 0 to 10). Total = 1 + sum from k=1 to 10 of 8k. Step 3: Evaluate sum from k=1 to 10 of 8k. This is 8 times the sum from k=1 to 10 of k. The sum of the first 10 positive integers is 10*11/2 = 55. So 8 * 55 = 440. Step 4: Add the central tile: 440 + 1 = 441. Step 5: The total number of tiles is 441.

2. A geometric pattern is formed by stacking squares in a triangular arrangement. The first row has 1 square, the second row has 3 squares, the third row has 5 squares, and this pattern continues. The side length of each square is 2 cm. Write the total area of the pattern after n rows using sigma notation, then evaluate the total area for n = 10 rows. Answer: 400 cm² Solution: Row 1: 1 square Row 2: 3 squares Row 3: 5 squares This pattern continues.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the pattern of squares per row** The problem says: Row 1: 1 square Row 2: 3 squares Row 3: 5 squares This pattern continues. So the number of squares in row \( k \) is: \[ \text{Squares in row } k = 1 + (k-1) \times 2 \] Check: For \( k = 1 \): \( 1 + 0 = 1 \) For \( k = 2 \): \( 1 + 2 = 3 \) For \( k = 3 \): \( 1 + 4 = 5 \) Yes, so general formula: \( 2k - 1 \) squares in row \( k \). --- **Step 2: Area of one square** Each square has side length \( 2 \) cm. Area of one square = \( 2 \times 2 = 4 \) cm². --- **Step 3: Total area after \( n \) rows** Each row \( k \) has \( 2k - 1 \) squares, each of area \( 4 \) cm². So area of row \( k \) = \( (2k - 1) \times 4 \) cm². Total area after \( n \) rows: \[ \text{Total area} = \sum_{k=1}^{n} [ (2k - 1) \times 4 ] \] We can factor out the 4: \[ \text{Total area} = 4 \sum_{k=1}^{n} (2k - 1) \] --- **Step 4: Simplify the sum** We know: \[ \sum_{k=1}^{n} (2k - 1) = 2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \] \[ = 2 \times \frac{n(n+1)}{2} - n \] \[ = n(n+1) - n \] \[ = n^2 + n - n \] \[ = n^2 \] So the sum of the first \( n \) odd numbers is \( n^2 \). --- **Step 5: Total area formula** \[ \text{Total area} = 4 \times n^2 \] --- **Step 6: Evaluate for \( n = 10 \)** \[ \text{Total area} = 4 \times 10^2 = 4 \times 100 = 400 \text{ cm}^2 \] --- **Final Answer:** 400 cm²

3. Σ(k=0 to 5) (5k + 10) = ? Answer: 135 Solution: Number of terms: n = 5 - 0 + 1 = 6. First term (k=0): 5(0) + 10 = 0 + 10 = 10. Last term (k=5): 5(5) + 10 = 25 + 10 = 35.
   Full step-by-step solution

   Step 1: Number of terms: n = 5 - 0 + 1 = 6. Step 2: First term (k=0): 5(0) + 10 = 0 + 10 = 10. Step 3: Last term (k=5): 5(5) + 10 = 25 + 10 = 35. Step 4: Sum formula: S = n/2 * (first term + last term). Step 5: S = 6/2 * (10 + 35) = 3 * 45 = 135. The answer is 135.

4. A particle moves along a straight line with velocity given by v(t) = 3t² - 2t + 5 meters per second, where t is time in seconds. The total distance traveled during the first 4 seconds can be expressed as ∑ from k=1 to n of [3((k-1)Δt)² - 2((k-1)Δt) + 5]Δt, where Δt = 4/n. Evaluate this sum as n approaches infinity by first expressing it in closed form using sigma notation properties, then taking the limit. Answer: 68 Solution: Recognize that as n approaches infinity, the sum becomes the definite integral of v(t) from 0 to 4.
   Full step-by-step solution

   Step 1: Recognize that as n approaches infinity, the sum becomes the definite integral of v(t) from 0 to 4. Step 2: Set up the integral: ∫ from 0 to 4 of (3t² - 2t + 5) dt Step 3: Find the antiderivative: t³ - t² + 5t Step 4: Evaluate from 0 to 4: [4³ - 4² + 5×4] - [0³ - 0² + 5×0] Step 5: Calculate: [64 - 16 + 20] - [0] = 68 Step 6: The total distance traveled is 68 meters. The answer is 68.

5. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration C(t) in mg/L is given by the function C(t) = 5te^(-0.2t), 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 area under the concentration curve. Express this total exposure using sigma notation with 5 equal subintervals, then write the definite integral that would give the exact value. Answer: ∫ from 0 to 10 of 5te^(-0.2t) dt Solution: C(t) = 5 * t * e^(-0.2 * t) We want the total drug exposure over the first 10 hours, which is the area under the curve from t = 0 to t = 10.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understanding the problem** We are given the concentration function: C(t) = 5 * t * e^(-0.2 * t) We want the total drug exposure over the first 10 hours, which is the area under the curve from t = 0 to t = 10. --- **Step 2: Sigma notation for 5 equal subintervals** If we use 5 equal subintervals over [0, 10], then: - Width of each subinterval: Δt = (10 - 0) / 5 = 2 hours. - We can use right endpoints, left endpoints, or midpoints. The problem doesn't specify, so let's use right endpoints for definiteness. Right endpoints: t_i = 0 + i * Δt = 2i, for i = 1, 2, 3, 4, 5. So t_1 = 2, t_2 = 4, t_3 = 6, t_4 = 8, t_5 = 10. The Riemann sum in sigma notation is: Total exposure ≈ Σ (from i=1 to 5) [ C(t_i) * Δt ] = Σ (i=1 to 5) [ 5 * (2i) * e^(-0.2 * (2i)) * 2 ] = Σ (i=1 to 5) [ 20i * e^(-0.4i) ] So in sigma notation: Σ_{i=1}^{5} 20 i e^(-0.4 i) --- **Step 3: Exact value as a definite integral** The exact total exposure is the limit of such Riemann sums as n → ∞, which is the definite integral of C(t) from 0 to 10. So exact exposure = ∫ from 0 to 10 of C(t) dt = ∫ from 0 to 10 of 5 t e^(-0.2 t) dt --- **Step 4: Final answer** The problem asks to express total exposure using sigma notation with 5 equal subintervals, then write the definite integral for the exact value. We have: Sigma notation: Σ_{i=1}^{5} 20 i e^(-0.4 i) Definite integral: ∫_0^10 5 t e^(-0.2 t) dt Since the correct answer given is the integral form, that is the final exact representation. --- **Final Answer:** ∫ from 0 to 10 of 5 t e^(-0.2 t) dt

6. A triangular arrangement of dots is constructed. The first row has 1 dot, the second row has 3 dots, the third row has 5 dots, and this pattern continues. Each dot has a side length of 1 unit, so the number of dots in each row equals the length of that row. Aroha notices that the total number of dots in the first n rows can be expressed using sigma notation. If the arrangement has 9 rows, write the series in sigma notation and evaluate the total number of dots. Answer: 81 Solution: Identify the pattern. The first row has 1 dot, the second row has 3 dots, the third row has 5 dots. This is an arithmetic sequence where each term increases by 2.
   Full step-by-step solution

   Step 1: Identify the pattern. The first row has 1 dot, the second row has 3 dots, the third row has 5 dots. This is an arithmetic sequence where each term increases by 2. For row k, the number of dots is 2k - 1. Step 2: Write the series in sigma notation for 9 rows. The total number of dots S = sum from k=1 to 9 of (2k - 1). Step 3: Evaluate the sum. We can split it: S = 2 * (sum from k=1 to 9 of k) - (sum from k=1 to 9 of 1). Step 4: Compute sum from k=1 to 9 of k = 9 * 10 / 2 = 45. Step 5: Compute sum from k=1 to 9 of 1 = 9. Step 6: Substitute: S = 2 * 45 - 9 = 90 - 9 = 81. Step 7: The total number of dots is 81.