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  • n3 -7n + 3 is divisible by 3, for all natural numbers n.

n^3 -7n + 3 is divisible by 3, for all natural numbers n.

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n^3 -7n + 3 is divisible by 3... given

Now, we’ll substitute different values for n,

P (0) = 0^3 -7 \times 0 + 3 = 3, is divisible by 3

P (1) = 1^3 -7 \times 1 + 3 = -3, is divisible by 3

P (2) = 2^3 -7 \times 2 + 3 = -3, is divisible by 3

Now, let us consider,

P (k) = k^3 -7k + 3 be divisible by 3

Thus, k^3 -7k + 3 = 3x

We also get that,

P (k+1) = (k+1)^3 -7(k+1) + 3 = k^3 + 3k^2 + 3k +1 -7k -7 + 3

                        = 3x + 3(k^2+k-2)is divisible by 3

Thus, P (k+1) is also true,

Hence, by mathematical induction,

For each natural no. n it is true that, P (n) = n^3 -7n + 3 is divisible by 3.
 

Posted by

infoexpert21

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