Use mathematical induction to prove the statement is true for all positive integers n.3 + 
+ 
+ . . . + 
= 4
What will be an ideal response?
Answers may vary. Possible answer:
First we show that the statement is true when n = 1.
For n = 1, we get 3 = 4
Since 4
= 4
= 3 , P1 is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is,
Pk: 3 + 
+ 
+ . . . + 
= 4
is assumed true.
On the basis of the assumption that Pk is true, we need to show that Pk+1 is true.
Pk+1: 3 + 
+ 
+ . . . + 
= 4
So we assume that 
is true and add the next term, 
to both sides of the equation.
3 + 
+ 
+ . . . + 
+ 
= 4
+ 
3 + 
+ 
+ . . . + 
+ 
= 4 - 
+ 
3 + 
+ 
+ . . . + 
+ 
= 4 - 
3 + 
+ 
+ . . . + 
+ 
= 4
The last equation says that Pk+1 is true if Pk is assumed to be true. Therefore, by the principle of mathematical induction, the statement 
is true for all natural numbers n.
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