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Statement -1 for all natural numbers n , `2.7^(n)+3.5^(n)-5` is divisible by 24. Statement -2 if f(x) is divisible by x, then `f(x+1)-f(x)` is divisible by `x+1,forall x in N`.A. Statement -1 is true , Statement -2 is true, Statement -2 is correct explanation for Statement -2B. Statement -1 is true , Statement -2 is true , Statement -2 is not correct explanation for Staement -2C. Statement -1 is true , Statement -2 is falseD. Statement -1 is false , Statement - 2 is true. |
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Answer» Let `P(n):2.7^(n)+3.5^(n)-5` Step I For `n=1`, `P(1):2.7^1+3.5^1-5` :24 is divisible by 24. Step II Assume P(k) is divisible by 24, then `P(k):2.7^k+3.5^k-5=24lambda, lambda` is positive integer. Step III For `n=k+1`, `P(k+1)-P(k)=(2.7^(k+1)+3.5^(k+1)-5)-(2.7^k+3.5^k-5)` `=2.7^k(7-1)+3.5^k(5-1)` `12(7^k+5^k)` =divisible by 24 `=24 mu , forall mu in I" "[ because 7^k+5^k "is a always divisible by"24]` `therefore P(k+1)=P(k)+24mu=24lambda+24mu` `=24(lambda+mu)` Hence , `P(k=1)` is divisible by 24. Hence , Statement -1 true and Statement -2 is false . |
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