Understanding the Problem: Average of Consecutive Odd Numbers

The question asks us to find the product of the first and fifth numbers in a sequence of five consecutive odd natural numbers whose average is 27. To solve this, we first need to identify these five numbers.

Finding the Consecutive Odd Natural Numbers

Consecutive odd natural numbers follow a pattern where each number is 2 greater than the previous one. If we let the first odd number be \(n\), the five consecutive odd natural numbers can be represented as:

• First number: \(n\)
• Second number: \(n+2\)
• Third number: \(n+4\)
• Fourth number: \(n+6\)
• Fifth number: \(n+8\)

The average of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers.

Given that the average of these five numbers is 27, we can set up the following equation:

$$ \text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} $$

$$ 27 = \frac{n + (n+2) + (n+4) + (n+6) + (n+8)}{5} $$

Let's simplify the sum in the numerator:

$$ \text{Sum} = n + n+2 + n+4 + n+6 + n+8 $$

$$ \text{Sum} = (n + n + n + n + n) + (2 + 4 + 6 + 8) $$

$$ \text{Sum} = 5n + 20 $$

Now, substitute this sum back into the average equation:

$$ 27 = \frac{5n + 20}{5} $$

To solve for \(n\), multiply both sides by 5:

$$ 27 \times 5 = 5n + 20 $$

$$ 135 = 5n + 20 $$

Subtract 20 from both sides:

$$ 135 - 20 = 5n $$

$$ 115 = 5n $$

Divide both sides by 5:

$$ n = \frac{115}{5} $$

$$ n = 23 $$

So, the first number is 23. Since they are consecutive odd natural numbers, the sequence is:

• First number: \(n = 23\)
• Second number: \(n+2 = 23+2 = 25\)
• Third number: \(n+4 = 23+4 = 27\)
• Fourth number: \(n+6 = 23+6 = 29\)
• Fifth number: \(n+8 = 23+8 = 31\)

The five consecutive odd natural numbers are 23, 25, 27, 29, and 31.

Alternatively, for consecutive numbers (arithmetic progression), the average is equal to the middle number. Since there are five numbers, the third number is the middle number. Thus, the third number is equal to the average, which is 27. If the third number is 27, the preceding odd number is 27 - 2 = 25, and the number before that is 25 - 2 = 23. The succeeding odd number is 27 + 2 = 29, and the number after that is 29 + 2 = 31. So the numbers are 23, 25, 27, 29, 31. This confirms our previous calculation.

Calculating the Product of the First and Fifth Numbers

The question asks for the product of the first and fifth numbers.

• First number = 23
• Fifth number = 31

Product = First number \(\times\) Fifth number

Product = \(23 \times 31\)

Let's calculate the product:

$$ 23 \times 31 = 23 \times (30 + 1) $$

$$ = (23 \times 30) + (23 \times 1) $$

$$ = 690 + 23 $$

$$ = 713 $$

The product of the first and fifth number is 713.

Conclusion

The five consecutive odd natural numbers are 23, 25, 27, 29, and 31. The product of the first number (23) and the fifth number (31) is 713.

Revision Table: Average and Consecutive Numbers

Additional Information: Natural Numbers and Odd Numbers

Natural Numbers: These are the positive integers starting from 1. {1, 2, 3, 4, ...}

Odd Numbers: These are integers that are not divisible by 2. When divided by 2, they leave a remainder of 1. Examples include 1, 3, 5, 7, 9, -1, -3, etc.

Consecutive: Means following each other in uninterrupted succession.

Understanding the properties of average and consecutive numbers is key to solving such problems quickly and efficiently.