Step 1: Identify the arithmetic sequence: a₁ = 2, d = 3 (common difference)

Step 2: Find how many terms (n): aₙ = a₁ + (n - 1)d 50 = 2 + (n - 1)(3) 50 = 2 + 3n - 3 51 = 3n n = 17

Step 3: Use sum formula for arithmetic series: Sₙ = n(a₁ + aₙ)/2

Step 4: Calculate: S₁₇ = 17(2 + 50)/2 S₁₇ = 17(52)/2 S₁₇ = 17(26) S₁₇ = 442

Answer: 442

This is an arithmetic series with:

• $a_1 = 2$
• $a_n = 14$
• $n = 5$ terms

Use the formula: $S_n = \frac{n(a_1 + a_n)}{2}$

$S_5 = \frac{5(2 + 14)}{2} = \frac{5(16)}{2} = \frac{80}{2} = 40$

Answer: $40$

Step 1: Understand the notation: Sum of (3k + 1) for k = 1, 2, 3, ..., 10

Step 2: Write out the first few terms: k=1: 3(1) + 1 = 4 k=2: 3(2) + 1 = 7 k=3: 3(3) + 1 = 10 ... This is arithmetic with a₁ = 4, d = 3

Step 3: Find the last term: k=10: 3(10) + 1 = 31

Step 4: Use arithmetic sum formula: S₁₀ = 10(4 + 31)/2 S₁₀ = 10(35)/2 S₁₀ = 175

Step 5: Alternative - split the sum: Σ(3k + 1) = Σ3k + Σ1 = 3Σk + 10 = 3(1+2+...+10) + 10 = 3(55) + 10 = 165 + 10 = 175 ✓

Answer: 175

This is a geometric series with:

• $a_1 = 3$
• $r = 2$
• $n = 6$

Use the formula: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$

$S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2}$ $= 3 \cdot \frac{1 - 64}{-1}$ $= 3 \cdot \frac{-63}{-1}$ $= 3 \cdot 63 = 189$

Answer: $189$

Step 1: Identify as geometric: a₁ = 1, r = 2

Step 2: Find n (how many terms): aₙ = a₁ · rⁿ⁻¹ 512 = 1 · 2ⁿ⁻¹ 512 = 2ⁿ⁻¹ 2⁹ = 2ⁿ⁻¹ n - 1 = 9 n = 10

Step 3: Use geometric sum formula: Sₙ = a₁(rⁿ - 1)/(r - 1)

Step 4: Calculate: S₁₀ = 1(2¹⁰ - 1)/(2 - 1) S₁₀ = (1024 - 1)/1 S₁₀ = 1023

Answer: 1023

Step 1: Write out the terms: k=0: 2⁰ = 1 k=1: 2¹ = 2 k=2: 2² = 4 k=3: 2³ = 8 k=4: 2⁴ = 16 k=5: 2⁵ = 32 k=6: 2⁶ = 64

Step 2: This is geometric series: a₁ = 1, r = 2, n = 7 terms (k goes from 0 to 6)

Step 3: Use sum formula: S₇ = 1(2⁷ - 1)/(2 - 1) S₇ = (128 - 1)/1 S₇ = 127

Step 4: Verify by adding: 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 ✓

Answer: 127

This is an infinite geometric series with:

• $a_1 = 8$
• $r = \frac{4}{8} = \frac{1}{2}$

Since $|r| = \frac{1}{2} < 1$, the series converges.

Use the formula: $S = \frac{a_1}{1 - r}$

$S = \frac{8}{1 - \frac{1}{2}}$ $= \frac{8}{\frac{1}{2}}$ $= 16$

Answer: $16$

Step 1: Identify the arithmetic sequence: a₁ = 15 (first row) d = 2 (each row has 2 more) n = 20 (total rows)

Step 2: Find the number of seats in the last row: a₂₀ = a₁ + (n - 1)d a₂₀ = 15 + (20 - 1)(2) a₂₀ = 15 + 38 a₂₀ = 53

Step 3: Find total seats using sum formula: Sₙ = n(a₁ + aₙ)/2 S₂₀ = 20(15 + 53)/2 S₂₀ = 20(68)/2 S₂₀ = 20(34) S₂₀ = 680

Step 4: Verify the pattern: Row 1: 15 Row 2: 17 Row 3: 19 ... Row 20: 53 This makes sense ✓

Answer: 680 seats