Question

Part b assume the statement is true for n = k. prove that it must be true for n = k + 1, therefore proving it true for all natural numbers, n. t hint: since the total number of dots increases by n each time, prove that d (k) + (k + 1) = d(k+1).

Answer 1

d( k+1) is only true when d(k) is true ,  d(1) should also be true for d(k+1) statement to be true.

What is Mathematical Induction ?

For each and every natural number n, mathematical induction is a method of demonstrating a statement, theorem, or formula that is presumed to be true.

It is given that

the statement is true for n = k

According to the principle of Mathematical Induction

Let d(n) be a statement involving Natural Number n such that

• $\rm d(1) \;is\; true$

• $\rm d(m) \;is\; true$

• $\rm d(m +1)\; is\; true$

So ,  the statement d(n) is true for all the n natural numbers.

According to the second principle of Mathematical Induction

Let d(n) be a statement involving Natural number n such that

• $\rm d(1) \;is\; true$

• $\rm d(m) \;is\; true$ when $\rm d(n) is true$ for all n where 1 ≤ n ≤m .

• Then the statement $\rm d(n) is true$ for all the values of n ( natural number)

Therefore , d( k+1) is only true when d(k) is true , also d(1) should also be true.

To know more about Mathematical Induction

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