Natural Numbers and Whole Numbers

46

Step 1: We are given the formula M = 3n + 1, where n is the figure number. Step 2: For the 15th figure, substitute n = 15 into the formula. Step 3: M = 3 × 15 + 1 = 45 + 1 = 46. Step 4: Verify the pattern: for n=1, M=4 (1 square needs 4 sticks); for n=2, M=7 (2 squares in a row need 7 sticks). The formula is correct. Answer = 46.

100

Step 1: The pattern states: Sum of first n odd natural numbers = n². Step 2: Here we need sum of first 10 odd numbers, so n = 10. Step 3: Sum = 10² = 10 × 10 = 100. Step 4: Verify by listing: 1+3+5+7+9+11+13+15+17+19 = 100. ✓ Common mistake: Students sometimes use the formula for natural numbers n(n+1)/2 = 55, which gives sum of first 10 natural numbers, not first 10 odd numbers.

₹6,480

Step 1: Total earnings = 324 × 15 + 324 × 5. Step 2: Use distributive property — take 324 as common: 324 × (15 + 5). Step 3: Add inside bracket: 15 + 5 = 20. Step 4: Multiply: 324 × 20 = 6,480. So total earnings = ₹6,480. Common mistake: Calculating 324 × 15 = 4,860 and 324 × 5 = 1,620 separately is correct too but using the property saves time.

Distributive property of multiplication over addition

Step 1: Look at what is happening: 56 is being multiplied by a sum (30 + 7). Step 2: The multiplication is distributed over each term of the addition: 56 × 30 and 56 × 7. Step 3: This is exactly the distributive property: a × (b + c) = a × b + a × c. Step 4: Commutative changes order (a×b = b×a), associative changes grouping ((a×b)×c = a×(b×c)), closure means result stays whole number. None of these match. The answer is Distributive Property.