Geometric Sequences

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Geometric Sequences · Grade 12 · Worksheet 3

1. Emma is a microbiologist studying the growth of a bacterial colony in a laboratory. She observes that the colony follows a geometric growth pattern. Initially, there are 7 bacterial cells. Every 4 hours, the number of cells triples. Emma wants to determine the number of bacterial cells present after 24 hours (the 7th term of the sequence) and the total number of cells that have existed over the entire 24-hour period (sum of the first 7 terms). Find the number of cells at 24 hours and the total number of cells over the first 7 time intervals. Answer: a7 = 5103, S7 = 7651 Solution: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 7 (since we start at time 0 and have intervals every 4 hours up to 24 hours, that gives 7 terms).
   Full step-by-step solution

   Step 1: Identify the first term a1 = 7 and the common ratio r = 3. The number of terms n = 7 (since we start at time 0 and have intervals every 4 hours up to 24 hours, that gives 7 terms). Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=7, a7 = 7 * 3^(7-1) = 7 * 3^6. Step 3: Calculate 3^6. 3^2 = 9, 3^4 = 9 * 9 = 81, 3^6 = 81 * 9 = 729. Step 4: Multiply by 7: a7 = 7 * 729 = 5103. Step 5: Use the sum formula S_n = a1 * (1 - r^n) / (1 - r). For n=7, S7 = 7 * (1 - 3^7) / (1 - 3). Step 6: Calculate 3^7 = 3 * 3^6 = 3 * 729 = 2187. Step 7: Numerator: 1 - 2187 = -2186. Denominator: 1 - 3 = -2. Step 8: (1 - r^n) / (1 - r) = (-2186) / (-2) = 1093. Step 9: Multiply by a1: S7 = 7 * 1093 = 7651. The number of cells at 24 hours is 5103 and the total number of cells over the first 7 time intervals is 7651.

2. Tane is constructing a visual pattern using a series of nested regular pentagons. The largest pentagon has a side length of 45 cm. Each subsequent pentagon is inscribed within the previous one by connecting the midpoints of its sides, forming a smaller rotated pentagon inside. The side lengths of these pentagons follow a geometric sequence. The common ratio for this pattern, derived from the geometry of connecting midpoints of a regular pentagon, is given by the golden ratio conjugate: r = (sqrt(5) - 1)/2. What is the side length of the 7th pentagon, and what is the sum of the perimeters of the first 7 pentagons? (Perimeter of a regular pentagon = 5 × side length) Answer: Side length of 7th pentagon = 45 * ((sqrt(5) - 1)/2)^6 cm; Sum of perimeters = 225 * (1 - ((sqrt(5) - 1)/2)^7) / (1 - (sqrt(5) - 1)/2) cm Solution: Identify the common ratio. The problem states that the common ratio for this geometric sequence is r = (sqrt(5) - 1)/2. Find the side length of the 7th pentagon.
   Full step-by-step solution

   Step 1: Identify the common ratio. The problem states that the common ratio for this geometric sequence is r = (sqrt(5) - 1)/2. Step 2: Find the side length of the 7th pentagon. The first term a1 = 45 cm. Using the formula for the nth term of a geometric sequence, a_n = a1 * r^(n-1): a_7 = 45 * ((sqrt(5) - 1)/2)^(7-1) a_7 = 45 * ((sqrt(5) - 1)/2)^6 cm. Step 3: Find the perimeter of the first pentagon. Perimeter P1 = 5 * 45 = 225 cm. Step 4: Sum of perimeters of the first 7 pentagons. The perimeters form a geometric sequence with first term P1 = 225 and common ratio r = (sqrt(5) - 1)/2. Using the sum formula for a geometric series, S_n = a1 * (1 - r^n) / (1 - r): S_7 = 225 * (1 - ((sqrt(5) - 1)/2)^7) / (1 - (sqrt(5) - 1)/2). Step 5: Simplify the denominator: 1 - (sqrt(5) - 1)/2 = (2 - sqrt(5) + 1)/2 = (3 - sqrt(5))/2. So the sum formula becomes: S_7 = 225 * (1 - ((sqrt(5) - 1)/2)^7) / ((3 - sqrt(5))/2) S_7 = (450 * (1 - ((sqrt(5) - 1)/2)^7)) / (3 - sqrt(5)). This expression can be rationalized, but the answer is typically left in the simplified form below. The answer is: Side length of 7th pentagon = 45 * ((sqrt(5) - 1)/2)^6 cm, Sum of perimeters = 225 * (1 - ((sqrt(5) - 1)/2)^7) / (1 - (sqrt(5) - 1)/2) cm.

3. Mere is a marine biologist studying the population growth of a rare species of sea star in a protected marine reserve. She observes that the number of sea stars in a specific quadrat follows a geometric pattern. In the first year of her study, there are 8 sea stars. Each subsequent year, the population increases by a factor of 4 (the number of sea stars quadruples each year). Mere wants to know the population size after 6 years (the 6th term of the sequence) and the total number of sea stars that have been present in the quadrat over the first 6 years. Find the population in the 6th year and the total population over the first 6 years. Answer: a6 = 32768, S6 = 43688 Solution: Identify the first term a1 = 8 and the common ratio r = 4. The number of terms n = 6. Use the nth term formula a_n = a1 * r^(n-1).
   Full step-by-step solution

   Step 1: Identify the first term a1 = 8 and the common ratio r = 4. The number of terms n = 6. Step 2: Use the nth term formula a_n = a1 * r^(n-1). For n=6, a6 = 8 * 4^(6-1) = 8 * 4^5. Step 3: Calculate 4^5 = 4 * 4 * 4 * 4 * 4 = 1024. Step 4: Multiply by 8: a6 = 8 * 1024 = 8192. Step 5: Use the sum formula S_n = a1 * (r^n - 1) / (r - 1). For n=6, S6 = 8 * (4^6 - 1) / (4 - 1). Step 6: Calculate 4^6 = 4 * 4^5 = 4 * 1024 = 4096. Step 7: Numerator: 4096 - 1 = 4095. Denominator: 4 - 1 = 3. Step 8: Divide: 4095 / 3 = 1365. Step 9: Multiply by a1: S6 = 8 * 1365 = 10920. The population in the 6th year is 8192 sea stars and the total population over the first 6 years is 10920 sea stars.

4. ∑(n=1 to 5) 7 × (-3)^(n-1) = ? Answer: 427 Solution: a₁ = 7 × (-3)^(1-1) = 7 × (-3)^0 = 7 × 1 = 7 r = -3 Sₙ = a₁ × (1 - rⁿ) / (1 - r) Substitute n = 5, a₁ = 7, r = -3 S₅ = 7 × (1 - (-3)⁵) / (1 - (-3)) Calculate (-3)⁵ (-3)⁵ = -243 S₅ = 7 × (1 - (-243)) / (1 + 3) S₅ = 7 × (1 + 243) / 4 S₅ = 7 × 244 / 4 S₅ = 1708 / 4 S₅ = 427 Term 1: 7 × (-3)^0 = 7…
   Full step-by-step solution

   Step 1: Identify the first term (a₁) and common ratio (r) a₁ = 7 × (-3)^(1-1) = 7 × (-3)^0 = 7 × 1 = 7 r = -3 Step 2: Use the geometric series sum formula for n terms Sₙ = a₁ × (1 - rⁿ) / (1 - r) Step 3: Substitute n = 5, a₁ = 7, r = -3 S₅ = 7 × (1 - (-3)⁵) / (1 - (-3)) Step 4: Calculate (-3)⁵ (-3)⁵ = -243 Step 5: Substitute into the formula S₅ = 7 × (1 - (-243)) / (1 + 3) S₅ = 7 × (1 + 243) / 4 S₅ = 7 × 244 / 4 S₅ = 1708 / 4 S₅ = 427 Step 6: Verify by calculating individual terms Term 1: 7 × (-3)^0 = 7 Term 2: 7 × (-3)^1 = -21 Term 3: 7 × (-3)^2 = 63 Term 4: 7 × (-3)^3 = -189 Term 5: 7 × (-3)^4 = 567 Sum = 7 + (-21) + 63 + (-189) + 567 = -14 + 63 + (-189) + 567 = 49 + (-189) + 567 = -140 + 567 = 427 The answer is 427.

5. ∑(n=1 to 5) 5 × (-4)^(n-1) = ? Answer: 1025 Solution: a₁ = 5 × (-4)^(1-1) = 5 × (-4)^0 = 5 × 1 = 5 r = -4 Sₙ = a₁ × (1 - rⁿ) / (1 - r) Substitute n = 5, a₁ = 5, r = -4 S₅ = 5 × (1 - (-4)⁵) / (1 - (-4)) Calculate (-4)⁵ (-4)⁵ = -1024 S₅ = 5 × (1 - (-1024)) / (1 + 4) S₅ = 5 × (1 + 1024) / 5 S₅ = 5 × 1025 / 5 S₅ = 1025 The answer is 1025.
   Full step-by-step solution

   Step 1: Identify the first term (a₁) and common ratio (r) a₁ = 5 × (-4)^(1-1) = 5 × (-4)^0 = 5 × 1 = 5 r = -4 Step 2: Use the geometric series sum formula for n terms Sₙ = a₁ × (1 - rⁿ) / (1 - r) Step 3: Substitute n = 5, a₁ = 5, r = -4 S₅ = 5 × (1 - (-4)⁵) / (1 - (-4)) Step 4: Calculate (-4)⁵ (-4)⁵ = -1024 Step 5: Substitute into the formula S₅ = 5 × (1 - (-1024)) / (1 + 4) S₅ = 5 × (1 + 1024) / 5 S₅ = 5 × 1025 / 5 S₅ = 1025 The answer is 1025.

6. ∑(n=1 to 5) 4 × (1/2)^(n-1) = ? Answer: 7.75 Solution: Identify the first term a₁ = 4 and common ratio r = 1/2 Use the finite geometric series formula S_n = a₁(1 - rⁿ)/(1 - r) Substitute n = 5: S₅ = 4(1 - (1/2)⁵)/(1 - 1/2) Calculate (1/2)⁵ = 1/32 = 0.03125 Calculate 1 - 0.03125 = 0.96875 Calculate 1 - 1/2 = 1/2 = 0.5 S₅ = 4 × 0.96875 / 0.5 4 ×…
   Full step-by-step solution

   Step 1: Identify the first term a₁ = 4 and common ratio r = 1/2 Step 2: Use the finite geometric series formula S_n = a₁(1 - rⁿ)/(1 - r) Step 3: Substitute n = 5: S₅ = 4(1 - (1/2)⁵)/(1 - 1/2) Step 4: Calculate (1/2)⁵ = 1/32 = 0.03125 Step 5: Calculate 1 - 0.03125 = 0.96875 Step 6: Calculate 1 - 1/2 = 1/2 = 0.5 Step 7: S₅ = 4 × 0.96875 / 0.5 Step 8: 4 × 0.96875 = 3.875 Step 9: 3.875 ÷ 0.5 = 7.75 The answer is 7.75.