Join the IDNLearn.com community and get your questions answered by knowledgeable individuals. Join our interactive community and get comprehensive, reliable answers to all your questions.

Assume the statement is true for [tex]n = k[/tex]. Prove that it must be true for [tex]n = k+1[/tex], thereby proving it true for all natural numbers [tex]n[/tex].

Hint: Since the total number of dots increases by [tex]n[/tex] each time, prove that [tex]d(k) + (k+1) = d(k+1)[/tex].


Sagot :

Let's work through the problem step-by-step.

1. Base Case: Assume the statement is true for some [tex]\( n = k \)[/tex].

Let [tex]\( d(k) \)[/tex] represent the number of dots when [tex]\( n = k \)[/tex].

2. Inductive Step: We need to prove that if the statement holds for [tex]\( n = k \)[/tex], then it must also hold for [tex]\( n = k+1 \)[/tex].

According to our assumption, [tex]\( d(k) \)[/tex] is the total number of dots when [tex]\( n = k \)[/tex].

3. Incrementing [tex]\( n \)[/tex]: Suppose we increase [tex]\( n \)[/tex] by 1, that is, we examine the case [tex]\( n = k+1 \)[/tex].

4. Expression for [tex]\( n = k+1 \)[/tex]: When [tex]\( n \)[/tex] becomes [tex]\( k + 1 \)[/tex], it means we are adding an additional level with [tex]\( k + 1 \)[/tex] dots to our total dot count.

5. Inductive Hypothesis: We assume
[tex]\[ d(k+1) = d(k) + (k + 1) \][/tex]
Here, [tex]\( d(k) \)[/tex] is the number of dots if [tex]\( n = k \)[/tex] and adding [tex]\( k + 1 \)[/tex] dots for the next level.

6. Combining Information:
Let’s denote the number of dots at [tex]\( n = k+1 \)[/tex] by [tex]\( d(k+1) \)[/tex]. According to our induction step, we have:
[tex]\[ d(k+1) = d(k) + (k + 1) \][/tex]

This shows that the number of dots for [tex]\( n = k+1 \)[/tex] is built on the number of dots for [tex]\( n = k \)[/tex] by adding [tex]\( k+1 \)[/tex] more dots.

7. Conclusion:
We’ve shown that if the statement holds for [tex]\( n = k \)[/tex], i.e., [tex]\( d(k) \)[/tex] dots, then it also holds for [tex]\( n = k+1 \)[/tex], i.e., [tex]\( d(k+1) = d(k) + (k + 1) \)[/tex].

By mathematical induction, since the statement is true for [tex]\( n=k \)[/tex] and leads directly to being true for [tex]\( n=k+1 \)[/tex], we conclude that the statement is true for all natural numbers [tex]\( n \)[/tex].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.