**Step 1: Complete the square for the expression n² - 30n + 236.**
To complete the square for n² - 30n, we take half of the coefficient of n (-30), which is -15, and square it.
The expression becomes:
(n - 15)² - (-15)² + 236
**Step 2: Simplify the expression.**
= (n - 15)² - 225 + 236
= (n - 15)² + 11
**Step 3: Analyse the completed square form to find the minimum value.**
The term (n - 15)² is a squared number, so its minimum possible value is 0. This occurs when n = 15.
For any integer value of n, (n - 15)² ≥ 0.
Therefore, the minimum value of the entire expression (n - 15)² + 11 will occur when (n - 15)² is at its minimum.
Minimum value = 0 + 11 = 11.
**Step 4: Conclude based on the minimum value.**
The minimum value of any term in the sequence is 11.
Since 11 is a two-digit number, and all other terms will be greater than 11, all terms of the sequence must have two or more digits.
For example, if n=1, the term is (1-15)² + 11 = (-14)² + 11 = 196 + 11 = 207.
If n=15, the term is (15-15)² + 11 = 0 + 11 = 11.
We are given the nth term as Tₙ = n² - 30n + 236.
We need to complete the square for this expression.
1. **Halve the coefficient of n:** The coefficient of n is -30. Half of -30 is -15.
2. **Write the squared bracket:** This gives us (n - 15)².
3. **Expand this bracket to see what we have:** (n - 15)² = n² - 30n + 225.
4. **Adjust the constant:** Our original expression has +236, but the bracket gives +225. To get from 225 to 236, we need to add 11.
So, n² - 30n + 236 = (n² - 30n + 225) + 11 = (n - 15)² + 11.
The nth term of the sequence is given by (n - 15)² + 11.
Now we need to show that all terms have two or more digits. We can do this by finding the minimum value of the expression.
- The term (n - 15)² is a square, so its value is always greater than or equal to 0 for any integer n.
- The minimum value of (n - 15)² is 0, which occurs when n = 15.
- Therefore, the minimum value of the entire expression is 0 + 11 = 11.
Since the smallest possible value of any term in the sequence is 11, and all other terms will be larger, every term in the sequence is a positive integer greater than or equal to 11. All such numbers have two or more digits.
Expert