Question

If a statement p(n) is true for n= k+1 , then can we conclude that it is true for n = k ?

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Answer 1

Answer:

Yes P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of Mathematical Induction, P(n) is true for all n ∈ N

Step-by-step explanation:

If the statement is true for n = k (where k is a particular but arbitrary natural

number), then the statement is also true for n = k + 1, i.e, truth of P(k) implies

the truth of P(k + 1). Then P(n) is true for all natural numbers n.

Example 1 1 + 3 + 5 + ... + (2n – 1) = n^2

Solution : Let the given statement P(n) be defined as P(n) : 1 + 3 + 5 +...+ (2n –1) = n^2

, for n ∈ N. Note that P(1) is true, since

P(1) : 1 = 1^2

Assume that P(k) is true for some k ∈ N, i.e.,

P(k) : 1 + 3 + 5 + ... + (2k – 1) = k^2

Now, to prove that P(k + 1) is true, we have

1 + 3 + 5 + ... + (2k – 1) + (2k + 1)

= k^ 2 + (2k + 1)

= k ^2 + 2k + 1 = (k + 1)^2

Thus, P(k + 1) is true, whenever P(k) is true.

Thus, P(k + 1) is true, whenever P(k) is true. Hence, by the Principle of Mathematical Induction, P(n) is true for all n ∈ N

Hope it works....

Answer 2

Answer :-

We are provided with a statement which says

p(n) is true for n = k +1

Now it is asking can we conclude that

p(n) is true for n = k

Well first we should go through the Principle of induction.

First Principle of Mathematical Induction

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

i⟩ p(1) is true and

ii⟩ p(m) is true and

iii⟩ p(m + 1) is true

Then the statement p(n) is true for all the n Natural numbers.

Second Principle of Mathematical Induction

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

i⟩ p(1) is true and

ii⟩ p(m + 1) is true whenever p(n) is true for all n where 1 ≤ n ≤ m

Then the statement p(n) is true for all values of n ( Natural numbers)

Now we are able to see

p(m + 1) is only true when p(m) is true and p(1) is true .

So we can say that

p(k + 1) is only true when p(k) is true

also p(1) should be true.

So when p(k + 1) is true then we can say that p(k) is true.

Now we generalise it as when

p(k) and p(k + 1) is true then p(n) is true when n belongs to Natural numbers.

Note that :- There are some sequence where the situation is defined ,, there the p(1) is the can be written as p(a) where a is the first term according to the given situation.