Concept explainers
To prove:
The formula for all natural number using the principle of induction.
Explanation of Solution
Given info.
If the statement is true for and
When a statement is true for a natural number then it will also be true for its successor .
Then statement will be true for all natural number n. this is called principle of mathematical induction.
Proof:
Show that is true
Left hand side
R.H.S.
Thus, the statement is true for
Now, follow induction step.
If is true for then it will also be true for
Let us assume that is true condition that means
Then by taking above statement true, proceed and prove that its successor will also be true.
L.H.S.
As we know that
Substitute the value
Take as common factor
This is the value of R.H.S.
Now, we have full-filled both conditions of the principle of mathematical induction. The formula is therefore true for every natural number.
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