Module 6: Practice Problems

Step 1: Recall the formula for the nth term
For an arithmetic sequence: a_n = a₁ + (n - 1)d

Step 2: Find each term using the formula
a₁ = 7 (given)
a₂ = 7 + (2 - 1)(4) = 7 + 4 = 11
a₃ = 7 + (3 - 1)(4) = 7 + 8 = 15
a₄ = 7 + (4 - 1)(4) = 7 + 12 = 19
a₅ = 7 + (5 - 1)(4) = 7 + 16 = 23

Step 3: Verify the common difference
11 - 7 = 4, 15 - 11 = 4, 19 - 15 = 4, 23 - 19 = 4 (Check)

Step 1: Identify the first term and common difference
a₁ = 3
d = 8 - 3 = 5

Step 2: Use the formula a_n = a₁ + (n - 1)d
a_n = 3 + (n - 1)(5)

Step 3: Simplify the formula
a_n = 3 + 5n - 5
a_n = 5n - 2

Step 4: Verify with known terms
n = 1: a₁ = 5(1) - 2 = 3 (Check)
n = 2: a₂ = 5(2) - 2 = 8 (Check)
n = 3: a₃ = 5(3) - 2 = 13 (Check)

Step 1: Identify a₁ and d
a₁ = -5
d = -2 - (-5) = 3

Step 2: Use the formula a_n = a₁ + (n - 1)d
a₂₀ = -5 + (20 - 1)(3)

Step 3: Calculate
a₂₀ = -5 + 19(3)
a₂₀ = -5 + 57
a₂₀ = 52

Step 1: Identify known values
a₁ = 4
d = 10 - 4 = 6
a_n = 100 (last term)

Step 2: Use the formula a_n = a₁ + (n - 1)d
100 = 4 + (n - 1)(6)

Step 3: Solve for n
100 = 4 + 6n - 6
100 = 6n - 2
102 = 6n
n = 17

Step 4: Verify
a₁₇ = 4 + (17 - 1)(6) = 4 + 96 = 100 (Check)

Step 1: Identify a₁ and d
a₁ = 6
d = 11 - 6 = 5

Step 2: Find a₁₅
a₁₅ = 6 + (15 - 1)(5)
a₁₅ = 6 + 70 = 76

Step 3: Use the sum formula S_n = n(a₁ + a_n)/2
S₁₅ = 15(6 + 76)/2

Step 4: Calculate
S₁₅ = 15(82)/2
S₁₅ = 1230/2
S₁₅ = 615

Step 1: Recall the formula for the nth term
For a geometric sequence: a_n = a₁ · r^n-1

Step 2: Find each term
a₁ = 3 (given)
a₂ = 3 · 2¹ = 6
a₃ = 3 · 2² = 12
a₄ = 3 · 2³ = 24
a₅ = 3 · 2⁴ = 48

Step 3: Verify the common ratio
6/3 = 2, 12/6 = 2, 24/12 = 2, 48/24 = 2 (Check)

Step 1: Identify the first term and common ratio
a₁ = 5
r = 15/5 = 3

Step 2: Use the formula a_n = a₁ · r^n-1
a_n = 5 · 3^n-1

Step 3: Verify with known terms
n = 1: a₁ = 5 · 3⁰ = 5 (Check)
n = 2: a₂ = 5 · 3¹ = 15 (Check)
n = 3: a₃ = 5 · 3² = 45 (Check)

Step 1: Identify a₁ and r
a₁ = 64
r = 32/64 = 1/2

Step 2: Use the formula a_n = a₁ · r^n-1
a₈ = 64 · (1/2)⁷

Step 3: Calculate
a₈ = 64 · (1/128)
a₈ = 64/128
a₈ = 1/2

Step 1: Identify a₁ and r
a₁ = 2
r = 6/2 = 3

Step 2: Use the sum formula S_n = a₁(1 - rⁿ)/(1 - r)
S₆ = 2(1 - 3⁶)/(1 - 3)

Step 3: Calculate 3⁶
3⁶ = 729

Step 4: Complete the calculation
S₆ = 2(1 - 729)/(1 - 3)
S₆ = 2(-728)/(-2)
S₆ = -1456/(-2)
S₆ = 728

Step 1: Identify a₁ and r
a₁ = 12
r = 6/12 = 1/2

Step 2: Check if the series converges
Since |r| = |1/2| = 1/2 < 1, the series converges.

Step 3: Use the infinite sum formula S = a₁/(1 - r)
S = 12/(1 - 1/2)

Step 4: Calculate
S = 12/(1/2)
S = 12 · 2
S = 24

Step 1: Expand the summation by substituting each value of k
Σ(3k + 2) from k=1 to 5 = (3·1 + 2) + (3·2 + 2) + (3·3 + 2) + (3·4 + 2) + (3·5 + 2)

Step 2: Evaluate each term
= 5 + 8 + 11 + 14 + 17

Step 3: Add the terms
= 55

Step 1: Identify the pattern
Each term is a multiple of 4: 4(1), 4(2), 4(3), 4(4), 4(5), 4(6)

Step 2: Determine the general term
The kth term is 4k

Step 3: Determine the range
k goes from 1 to 6 (6 terms total)

Step 4: Write in summation notation
Σ(4k) from k=1 to 6

Step 1: Recall the formula for sum of squares
Σ(k²) from k=1 to n = n(n + 1)(2n + 1)/6

Step 2: Substitute n = 10
Σ(k²) from k=1 to 10 = 10(10 + 1)(2·10 + 1)/6

Step 3: Calculate
= 10(11)(21)/6
= 2310/6
= 385

Step 4: Verify by expansion (optional)
1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385 (Check)

Step 1: Expand the summation
Σ(2k² - k + 3) = Σ(2k²) - Σ(k) + Σ(3)

Step 2: Evaluate each summation separately
Σ(2k²) from k=1 to 4 = 2·Σ(k²) = 2(1 + 4 + 9 + 16) = 2(30) = 60
Σ(k) from k=1 to 4 = 1 + 2 + 3 + 4 = 10
Σ(3) from k=1 to 4 = 3 + 3 + 3 + 3 = 12

Step 3: Combine the results
60 - 10 + 12 = 62

Alternative: Direct expansion
k=1: 2(1) - 1 + 3 = 4
k=2: 2(4) - 2 + 3 = 9
k=3: 2(9) - 3 + 3 = 18
k=4: 2(16) - 4 + 3 = 31
Sum = 4 + 9 + 18 + 31 = 62 (Check)

Step 1: Identify the sequence type
This is an arithmetic sequence with a₁ = 50 and d = 25

Step 2: Find the 12th term (amount saved in month 12)
a₁₂ = 50 + (12 - 1)(25)
a₁₂ = 50 + 275 = 325

Step 3: Find the total saved using the sum formula
S₁₂ = 12(a₁ + a₁₂)/2
S₁₂ = 12(50 + 325)/2
S₁₂ = 12(375)/2
S₁₂ = 4500/2
S₁₂ = 2250

Step 1: Identify the sequence type
This is a geometric sequence with a₁ = 500 and r = 2

Step 2: Determine which term to find
After 8 hours means at the beginning of the 9th hour, so we need a₉

Step 3: Use the formula a_n = a₁ · r^n-1
a₉ = 500 · 2⁸

Step 4: Calculate
2⁸ = 256
a₉ = 500 · 256 = 128,000

Step 1: Identify the sequence type
This is a geometric sequence. The car retains 85% of its value each year.
a₁ = 30,000
r = 0.85 (100% - 15% = 85% = 0.85)

Step 2: Determine which term to find
After 5 years means the value at the end of year 5, which is a₆

Step 3: Use the formula a_n = a₁ · r^n-1
a₆ = 30,000 · (0.85)⁵

Step 4: Calculate
(0.85)⁵ = 0.4437...
a₆ = 30,000 · 0.4437
a₆ = 13,311.07

Step 1: Analyze the motion
Initial drop: 20 feet down
First bounce: up 20(0.6) = 12 feet, then down 12 feet
Second bounce: up 12(0.6) = 7.2 feet, then down 7.2 feet
And so on...

Step 2: Set up the total distance
Total = 20 + 2(12) + 2(7.2) + 2(4.32) + ...
Total = 20 + 2[12 + 7.2 + 4.32 + ...]
Total = 20 + 2[sum of infinite geometric series]

Step 3: Find the sum of the infinite geometric series
First term: a₁ = 12
Common ratio: r = 0.6
S = a₁/(1 - r) = 12/(1 - 0.6) = 12/0.4 = 30

Step 4: Calculate total distance
Total = 20 + 2(30) = 20 + 60 = 80 feet

Step 1: Identify the sequence type
This is an arithmetic sequence with a₁ = 15 and d = 3

Step 2: Find the number of seats in the 20th row
a₂₀ = 15 + (20 - 1)(3)
a₂₀ = 15 + 57 = 72

Step 3: Find the total number of seats
S₂₀ = 20(a₁ + a₂₀)/2
S₂₀ = 20(15 + 72)/2
S₂₀ = 20(87)/2
S₂₀ = 1740/2
S₂₀ = 870

Step 1: Identify the sequence type
This is a geometric sequence where each year the amount is multiplied by 1.06
a₁ = 5000
r = 1.06

Step 2: Determine which term to find
After 10 years means a₁₁ (initial amount is a₁)

Step 3: Use the formula a_n = a₁ · r^n-1
a₁₁ = 5000 · (1.06)¹⁰

Step 4: Calculate
(1.06)¹⁰ = 1.790847...
a₁₁ = 5000 · 1.790847
a₁₁ = 8954.24