Numbers and Sequences

590

Step 1: Identify a = 2, d = 5 - 2 = 3, n = 20. Step 2: Use S_n = n/2 × [2a + (n-1)d]. Step 3: S_20 = 20/2 × [2×2 + (20-1)×3]. Step 4: S_20 = 10 × [4 + 57] = 10 × 61 = 610. Wait — let me recheck: 2×2 = 4, (19)×3 = 57, 4+57 = 61, 10×61 = 610. Correction: The correct answer is 610 but listed as 590 above — recalculating carefully: a=2, d=3, n=20. S_20 = 10[4 + 57] = 10 × 61 = 610. So the correct answer is 610.

3

Step 1: We know t_n = ar^(n-1). For the 5th term: t_5 = ar^4. Step 2: Given t_5 = 48 and r = 2. So ar^4 = 48. Step 3: a × 2^4 = 48. Step 4: a × 16 = 48. Step 5: a = 48 ÷ 16 = 3. The first term is 3. Verify: 3, 6, 12, 24, 48 ✓ — the 5th term is indeed 48.

3

Step 1: The common ratio of a GP is found by dividing any term by the previous term. Step 2: r = t_2 / t_1 = 12 / 4 = 3. Step 3: Verify: t_3 / t_2 = 36 / 12 = 3. Step 4: t_4 / t_3 = 108 / 36 = 3. All ratios are equal, confirming r = 3. Common mistake: subtracting instead of dividing to find the ratio.

325

Step 1: Use the formula for sum of first n natural numbers: S = n(n+1)/2. Step 2: Here n = 25. Step 3: S = 25 × 26 / 2. Step 4: S = 650 / 2 = 325. Common mistake: using n(n+1) without dividing by 2, giving 650.