Questions
If then for every positive integer n is
detailed solution
Correct option is B
the result by the principle of mathematical induction on n.Step1 When n=1, by the definition of integral powers of a matrix, we have A1=A=1 1 11 1 11 1 1=31−1 31−1 31−131−1 31−1 31−131−1 31−1 31−1So, the result is true forn=1.Step2 Let the result be true forn=m .Then Am=3m−1 3m−1 3m−13m−1 3m−1 3m−13m−1 3m−1 3m−1 ...(i)Now we shall show that the result is true for n=m+1, i.e.n=m+1Am+1=3m 3m 3m3m 3m 3m3m 3m 3mBy the definition of integral powers of a matrix, we have Am+1=Am·A=3m-13m-13m-13m-13m-13m-13m-13m-13m-1111111111=3m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-13m-1+3m-1+3m-1=3.3m-13.3m-13.3m-13.3m-13.3m-13.3m-13.3m-13.3m-13.3m-1=3m3m3m3m3m3m3m3m3mThis shows that the result is true for, whenever it is true forn=m Hence, by the principle of mathematical induction the result is valid for any positive integer n.Similar Questions
The sum of series is
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