Use mathematical induction to prove the statement is true for all positive integers n.4 + 9 + 14 + . . . + (5n - 1) = (5n + 3)

Question

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Answer 1

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First we show that the statement is true when n = 1.
 For n = 1, we get 4 = (5(1) + 3) 
 Since (5(1) + 3) =  = 4, 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,
P_k:   is assumed true.
On the basis of the assumption that P_k is true, we need to show that P_k+1 is true. 
P_k+1: 
So we assume that  is true and add the next term,  to both sides of the equation.


The last equation says that P_k+1 is true if P_k is assumed to be true. Therefore, by the principle of mathematical induction, the statement 4 + 9 + 14 + ... + (5n - 1) =  (5n + 3) is true for all natural numbers n.