Arithmetic Sequences — Important Questions

2, 5, 8, 11, ..., 10, 7, 4, 1, ..., 1/2, 1, 3/2, 2, ...

Step 1: Check each sequence by finding consecutive differences. Option A: 5-2=3, 8-5=3, 11-8=3 (constant difference = 3) ✓. Option B: 4-1=3, 9-4=5, 16-9=7 (not constant) ✗. Option C: 7-10=-3, 4-7=-3, 1-4=-3 (constant difference = -3) ✓. Option D: 6-3=3, 12-6=6, 24-12=12 (not constant) ✗. Option E: 1-1/2=1/2, 3/2-1=1/2, 2-3/2=1/2 (constant difference = 1/2) ✓.

24

Step 1: In an arithmetic sequence, if the sum of positions is equal, the sum of terms is equal. Step 2: 4 + 8 = 12 and 5 + 7 = 12. Step 3: Since position sums are equal, a₄ + a₈ = a₅ + a₇ = 24. This property holds because terms equidistant from the center have the same sum.

The difference between consecutive terms is constant, The common difference can be negative, The sequence can have fractions as terms

Statement 1: TRUE - This is the definition of arithmetic sequence. Statement 2: TRUE - Example: 10, 7, 4, 1, ... has d = -3. Statement 3: FALSE - First term can be negative, e.g., -5, -2, 1, ... Statement 4: FALSE - Terms can be fractions, e.g., 1/2, 1, 3/2, ... Statement 5: TRUE - Arithmetic sequences can have fractional terms.

9

Step 1: Find common difference. Position difference = 7 - 3 = 4, Term difference = 27 - 15 = 12. Step 2: d = 12 ÷ 4 = 3. Step 3: Find first term using a₃ = 15. a₃ = a₁ + 2d, so 15 = a₁ + 2×3 = a₁ + 6. Step 4: a₁ = 15 - 6 = 9. Verification: Sequence is 9, 12, 15, 18, 21, 24, 27, ... ✓