Let's start by understanding the problem step by step.
To prove a statement for all natural numbers [tex]\( n \)[/tex], we will use a technique called mathematical induction. Here's how it works:
### Step 1: Base Case
We need to show that the statement is true for the initial value, typically [tex]\( n = 1 \)[/tex].
### Step 2: Inductive Step
Assume that the statement is true for some arbitrary natural number [tex]\( n = k \)[/tex]. This assumption is called the inductive hypothesis.
Then, we need to prove that the statement is true for [tex]\( n = k + 1 \)[/tex].
Now, let’s apply this to the given problem.
### Base Case
For [tex]\( n = 1 \)[/tex]:
- The total number of dots, denoted as [tex]\( d(1) \)[/tex], should be equal to 1.
This is trivially true as a single step consisting of 1 dot.
### Inductive Step
Assume the statement is true for [tex]\( n = k \)[/tex]. That means:
[tex]\[ d(k) = \frac{k(k + 1)}{2} \][/tex]
We need to prove that the statement is true for [tex]\( n = k + 1 \)[/tex]. In other words, we need to prove:
[tex]\[ d(k + 1) = \frac{(k + 1)(k + 2)}{2} \][/tex]
According to the hint, since the total number of dots increases by [tex]\( n \)[/tex] each time, we can express the relationship as:
[tex]\[ d(k + 1) = d(k) + (k + 1) \][/tex]
Using our inductive hypothesis [tex]\( d(k) = \frac{k(k + 1)}{2} \)[/tex], let's perform the calculations:
1. Start with the inductive hypothesis and add [tex]\( (k + 1) \)[/tex]:
[tex]\[ d(k + 1) = d(k) + (k + 1) \][/tex]
[tex]\[ d(k + 1) = \frac{k(k + 1)}{2} + (k + 1) \][/tex]
2. Factor out [tex]\( (k + 1) \)[/tex]:
[tex]\[ d(k + 1) = \frac{k(k + 1)}{2} + \frac{2(k + 1)}{2} \][/tex]
[tex]\[ d(k + 1) = \frac{k(k + 1) + 2(k + 1)}{2} \][/tex]
[tex]\[ d(k + 1) = \frac{(k + 1)(k + 2)}{2} \][/tex]
This matches the formula we aimed to prove for [tex]\( d(k + 1) \)[/tex].
### Conclusion
By mathematical induction, we have shown that if the statement is true for [tex]\( n = k \)[/tex], it is also true for [tex]\( n = k + 1 \)[/tex]. Since the base case [tex]\( n = 1 \)[/tex] is true and the inductive step is proven, we can conclude that the statement is true for all natural numbers [tex]\( n \)[/tex].
Thus, we have proved that the formula [tex]\( d(n) = \frac{n(n + 1)}{2} \)[/tex] correctly represents the total number of dots for any natural number [tex]\( n \)[/tex].
