Let's solve the problem step by step.
### Step (i): Expressing the Second Number
First, let's denote the two consecutive natural numbers as \(\( x \)\) and \(\( x + 1 \)\).
### Step (ii): Finding the Numbers
We know that the product of these two numbers is 30:
\(\[ x \times (x + 1) = 30 \]\)
This equation can be expanded and rewritten as:
\(\[ x^2 + x - 30 = 0 \]\)
This is a quadratic equation in standard form (\(\( ax^2 + bx + c = 0 \)\)). We can solve it using the quadratic formula:
\(\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]\)
For our equation \(\( x^2 + x - 30 = 0 \)\), the coefficients are \(\( a = 1 \)\), \(\( b = 1 \)\), and \(\( c = -30 \)\).
Plug these values into the quadratic formula:
\(\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-30)}}{2 \cdot 1} \]\)
Simplifying inside the square root:
\(\[ x = \frac{-1 \pm \sqrt{1 + 120}}{2} \]\)
\(\[ x = \frac{-1 \pm \sqrt{121}}{2} \]\)
\(\[ x = \frac{-1 \pm 11}{2} \]\)
This gives us two solutions:
\(\[ x = \frac{-1 + 11}{2} = \frac{10}{2} = 5 \]\)
\(\[ x = \frac{-1 - 11}{2} = \frac{-12}{2} = -6 \]\)
Since \(\( x \)\) represents a natural number, we discard the negative solution. So, \(\( x = 5 \)\).
Therefore, the two consecutive natural numbers are:
\(\[ x = 5 \]\)
\(\[ x + 1 = 6 \]\)
### Step (iii): Finding the Number to Add to the Smaller Number
We need to determine the number \(\( z \)\) that should be added to the smaller number (which is 5) such that the sum of the squares of the two numbers becomes 85.
Let:
\(\[ 5 + z \]\)
\(\[ 6 \]\)
The sum of the squares of these two numbers should be equal to 85:
\(\[ (5 + z)^2 + 6^2 = 85 \]\)
Let's solve this equation step by step:
\(\[ (5 + z)^2 + 36 = 85 \]\)
\(\[ (5 + z)^2 = 85 - 36 \]\)
\(\[ (5 + z)^2 = 49 \]\)
Taking the square root of both sides:
\(\[ 5 + z = \pm \sqrt{49} \]\)
\(\[ 5 + z = \pm 7 \]\)
This gives us two solutions:
\(\[ 5 + z = 7 \]\)
\(\[ z = 2 \]\)
\(\[ 5 + z = -7 \]\)
\(\[ z = -12 \]\)
Since \(\( z \)\) represents an additive number and typically, we would consider only non-negative values for such contexts, we take:
\(\[ z = 2 \]\)
### Conclusion:
1. The two consecutive natural numbers are 5 and 6.
2. The number \(\( z \)\) that should be added to the smaller number (5) to make the sum of the squares of the two numbers 85 is \(\( z = 2 \)\).
So, the final answer is:
- The two consecutive natural numbers are 5 and 6.
- The number to be added to 5 is 2.
