Practice: Arithmetic Theorems

Answer: 55
Step-by-step Explanation:
1. The sum of the first n natural numbers is given by the formula n(n+1)/2.
2. Here, n = 10.
3. Substitute n into the formula: 10 × 11 / 2.
4. Calculate: 10 × 11 = 110.
5. Divide by 2: 110 / 2 = 55.
6. So, the sum is 55.

Answer: 400
Step-by-step Explanation:
1. The sum of the first n odd numbers is n².
2. Here, n = 20.
3. Calculate: 20 × 20 = 400.
4. So, the sum is 400.

Answer: 240
Step-by-step Explanation:
1. The sum of the first n even numbers is n(n+1).
2. Here, n = 15.
3. Substitute: 15 × 16 = 240.
4. So, the sum is 240.

Answer: 120
Step-by-step Explanation:
1. The product of the first n natural numbers is called factorial and written as n!
2. For n = 5: 5! = 1 × 2 × 3 × 4 × 5.
3. Calculate step by step: 1 × 2 = 2, 2 × 3 = 6, 6 × 4 = 24, 24 × 5 = 120.
4. So, the product is 120.

Answer: 30
Step-by-step Explanation:
1. The sum of the squares of the first n natural numbers is n(n+1)(2n+1)/6.
2. Here, n = 4.
3. Substitute: 4 × 5 × 9 / 6.
4. Calculate: 4 × 5 = 20, 20 × 9 = 180.
5. Divide by 6: 180 / 6 = 30.
6. So, the sum is 30.

Answer: 36
Step-by-step Explanation:
1. The sum of the cubes of the first n natural numbers is [n(n+1)/2]².
2. Here, n = 3.
3. Calculate n(n+1): 3 × 4 = 12.
4. Divide by 2: 12 / 2 = 6.
5. Square the result: 6 × 6 = 36.
6. So, the sum is 36.

Answer: 1
Step-by-step Explanation:
1. By definition, the factorial of 0 is 1.
2. This is a special rule in mathematics to make formulas work smoothly.
3. So, 0! = 1.

Answer: n(n+1)/2
Step-by-step Explanation:
1. The sum of the first n natural numbers is n(n+1)/2.
2. This formula comes from adding numbers in pairs: 1 + n, 2 + (n-1), etc.
3. Each pair adds up to (n+1), and there are n/2 pairs.
4. Multiply: n × (n+1) / 2.

Answer: n(n+1)(2n+1)/6
Step-by-step Explanation:
1. The sum of the squares of the first n natural numbers is n(n+1)(2n+1)/6.
2. This formula is used to quickly add up numbers like 1² + 2² + ... + n².
3. Substitute n into the formula to get the answer.

Answer: [n(n+1)/2]²
Step-by-step Explanation:
1. The sum of the cubes of the first n natural numbers is [n(n+1)/2]².
2. This means you first find the sum of the first n natural numbers, then square it.
3. So, add 1 + 2 + ... + n, then square the result.

Answer: 11
Step-by-step Explanation:
1. The sum of the first n odd numbers is n².
2. Set n² = 121.
3. Take the square root: n = 11.
4. So, n is 11.

Answer: 14
Step-by-step Explanation:
1. The sum of the first n even numbers is n(n+1).
2. Set n(n+1) = 210.
3. Try n = 14: 14 × 15 = 210.
4. So, n is 14.

Answer: 120 + 24 = 144
Step-by-step Explanation:
1. 5! means 1 × 2 × 3 × 4 × 5 = 120.
2. 4! means 1 × 2 × 3 × 4 = 24.
3. Add: 120 + 24 = 144.

Answer: 7
Step-by-step Explanation:
1. The sum of the first n natural numbers is n(n+1)/2.
2. Set n(n+1)/2 = 28.
3. Multiply both sides by 2: n(n+1) = 56.
4. Try n = 7: 7 × 8 = 56.
5. So, n is 7.

Answer: True
Step-by-step Explanation:
1. The nth odd number is 2n-1.
2. The sum is 1 + 3 + 5 + ... + (2n-1).
3. There are n terms.
4. The sum is n² (try with n = 3: 1 + 3 + 5 = 9 = 3²).

Answer: True
Step-by-step Explanation:
1. The nth even number is 2n.
2. The sum is 2 + 4 + 6 + ... + 2n.
3. Factor out 2: 2 × (1 + 2 + ... + n) = 2 × n(n+1)/2 = n(n+1).

Answer: 6
Step-by-step Explanation:
1. The sum of the squares of the first n natural numbers is n(n+1)(2n+1)/6.
2. Set n(n+1)(2n+1)/6 = 91.
3. Try n = 6: 6 × 7 × 13 / 6 = 91.
4. So, n is 6.

Answer: 6
Step-by-step Explanation:
1. The sum of the cubes of the first n natural numbers is [n(n+1)/2]².
2. Set [n(n+1)/2]² = 441.
3. Take the square root: n(n+1)/2 = 21.
4. Try n = 6: 6 × 7 / 2 = 21.
5. So, n is 6.

Answer: 42
Step-by-step Explanation:
1. 7! = 1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040.
2. 5! = 1 × 2 × 3 × 4 × 5 = 120.
3. Divide: 5040 / 120 = 42.

Answer: 11
Step-by-step Explanation:
1. The sum of the first n natural numbers is n(n+1)/2.
2. Set n(n+1)/2 = 66.
3. Multiply both sides by 2: n(n+1) = 132.
4. Try n = 11: 11 × 12 = 132.
5. So, n is 11.