NCERT Solutions Class 10 Maths Chapter 5 — Arithmetic Progressions

Solution:

Step 1: For an AP, the common difference 'd' must be constant for all consecutive terms.

Step 2: Option A: 4-2=2, 8-4=4. Not an AP.

Step 3: Option B: 4-1=3, 9-4=5. Not an AP.

Step 4: Option C: 6-3=3, 9-6=3, 12-9=3. The common difference is constant (d=3), so it is an AP.

Step 5: Option D: 1/3 - 1/2 = -1/6, 1/4 - 1/3 = -1/12. Not an AP.

Solution:

Step 1: Statement A is the correct formula for the n-th term of an AP.

Step 2: Statement B is correct; for example, 5, 5, 5, ... is an AP with d=0.

Step 3: Statement C is incorrect. The terms of an AP always increase if 'd' is positive. If 'd' is negative, the terms decrease. If 'd' is zero, the terms remain constant.

Step 4: Statement D is the definition of an AP, hence it is correct.

Solution:

Step 1: The given AP is 5, 8, 11, 14, ... . Here, the first term a = 5.

Step 2: The common difference d = 8 - 5 = 3.

Step 3: To find the 10th term using the formula a_n = a + (n-1)d, we substitute n=10, a=5, d=3.

Step 4: a_10 = 5 + (10-1) × 3 = 5 + 9 × 3 = 5 + 27 = 32. The student's calculation is correct, so there is no mistake in this specific problem.

Solution:

Step 1: The formula for the n-th term of an AP is a_n = a + (n-1)d.

Step 2: So, a_18 = a + (18-1)d = a + 17d.

Step 3: And a_13 = a + (13-1)d = a + 12d.

Step 4: Then, a_18 - a_13 = (a + 17d) - (a + 12d) = a + 17d - a - 12d = 5d.

Step 5: Given d = 5, the value is 5 × 5 = 25.

Solution:

Step 1: For the terms of an AP to be decreasing, the common difference 'd' must be negative.

Step 2: Option A: d=3 (positive), terms increase (e.g., 7, 10, 13...).

Step 3: Option B: d=5 (positive), terms increase (e.g., -2, 3, 8...).

Step 4: Option C: d=-2 (negative), terms decrease (e.g., 10, 8, 6...).

Step 5: Option D: d=0, terms remain constant (e.g., 0, 0, 0...).