Arithmetic Sequences and Algebra — Important Questions

2, 4, 6, 8, ..., 5, 5, 5, 5, ..., 10, 7, 4, 1, ...

Step 1: Check common difference. Option 1: d = 2 (constant) ✓. Option 2: differences are 3, 5, 7 (not constant) ✗. Option 3: d = 0 (constant) ✓. Option 4: ratios are 2 (geometric, not arithmetic) ✗. Option 5: d = -3 (constant) ✓.

6

Step 1: In algebraic form xn = an + b, the coefficient of n is the common difference. Step 2: Here xn = 6n - 4, so a = 6. Step 3: Therefore, common difference d = 6.

Common difference is constant, Algebraic form is xn = an + b, Sum formula is Sn = n/2(2a + (n-1)d)

Step 1: Check each property. Constant common difference is the defining property ✓. Step 2: Algebraic form xn = an + b is correct ✓. Step 3: Sum formula is standard ✓. Step 4: Doubling describes geometric sequences ✗.

10

Step 1: First term a = 1/2, common difference d = 1/2. Step 2: Use an = a + (n-1)d. Step 3: a20 = 1/2 + (20-1)×1/2 = 1/2 + 19/2 = 20/2 = 10.