if B, C ARE n ROWED SQUARE MATRICES AND IF A=B+C , BC=CB ,C^{2}=0 ,THEN SHOW THAT FOR EVERY n is the element ofN, A^{ n+1 } =B^{n}(B+(n+1)C).

Let P(*n*) be the statement given by

For *n* = 1, we have

Hence, the statement is true for *n* = 1.

Assume that the statement is true for *n* = *k*.

We shall now show that *P*(*k* + 1) is true. That is we need to show that

Now,

So, the statement is true for *n* = *k* + 1.

Thus, P(*k*) is true ÞP(*k* + 1) is true.

Hence, by the principle of mathematical induction P (*n*) is true for all *n* ÎN.

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