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Prove by mathematical induction that n^(5)and n have the same unit digit for any natural numbern. |
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Answer» Solution :We have to prove that `n^(5) - n` is divisible by 10. For `n = 1, 1^(5) - 1 = 0` is divisible by 10. Also, `n = 2 , 2^(5) - 2 = 30` is divisible by 10. Thus, `P(1)` and `P(2)` are true. `K^(5) - k = 10m"........"(1)` Now, `(k + 1)^(5) - (k +1)` `= k^(5) + 5K^(4) + 10k^(3) + 10 k^(2) + 5 k + 1 -k - 1` `= (k^(5) - k) + 5 k(k^(3) + 1) + 10 k^(3) + 10 k^(2)` `= 10 m + 10 k^(3) + 10 k^(2) + 5k(k^(3) + 1)` [Using (1)] Clearly, `k(k^(3)+1)` is EVEN for `k in N`. Thus, `P(k+1)` is true WHENEVER`P(k)` is true, So, by the principleof mathematical induction,`P(n)` is true for any natural number n. |
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