Question

Prove that for any positive integer number n , n 3 + 2 n is divisible by 3

Answer 1

Statement P (n) is defined by  

n 3 + 2 n is divisible by 3  

STEP 1: We first show that p (1) is true. Let n = 1 and calculate n 3 + 2n  

1 3 + 2(1) = 3  

3 is divisible by 3  

hence p (1) is true.  

STEP 2: We now assume that p (k) is true  

k 3 + 2 k is divisible by 3  

is equivalent to  

k 3 + 2 k = 3 M , where M is a positive integer.  

We now consider the algebraic expression (k + 1) 3 + 2 (k + 1); expand it and group like terms  

(k + 1) 3 + 2 (k + 1) = k 3 + 3 k 2 + 5 k + 3  

= [ k 3 + 2 k] + [3 k 2 + 3 k + 3]  

= 3 M + 3 [ k 2 + k + 1 ] = 3 [ M + k 2 + k + 1 ]  

Hence (k + 1) 3 + 2 (k + 1) is also divisible by 3 and therefore statement P(k + 1) is true.

Answer 2

Answer:

i don't know dear .. sry for that

Step-by-step explanation: