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प्रश्न
2.7n + 3.5n − 5 is divisible by 24 for all n ∈ N.
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उत्तर
Given expression
2.7n + 3.5n - 5
Proof – Let, P(n) = 2.7n + 3.5n - 5 is divisible by 24.
We note that P (n) is true when n = 1,
Since P (1) = 14 + 15 – 5 = 24 which is divisible by 24.
Assume that it is true for P (k).
i.e. P(k) = 2.7k + 3.5k - 5 = 24q ......(1) when q ∈ N
Now according to the mathematical induction principle, we have to prove it is also true for P (k + 1) whenever p (k) is true.
Now substitute in place of k, (k + 1) we have,
⇒ P(k + 1) = 2.7k+1 + 3.5k+1 - 5
⇒ 2.7.7k + 3.5.5k - 5
Now add and subtract by 3.7.5k - 7.5 we have,
⇒ 2.7.7k + 3.5.5k - 5 + 3.7.5k - 7.5 - (3.7.5k - 7.5)
⇒ 7(2.7k + 3.5k - 5) + 3.5.5k - 5 - (3.7.5k - 7.5)
Now from equation (1) we have,
⇒ 7(24q) + 3.5.5k - 5 - 3.7.5k + 7.5
⇒ 7(24q) - 2.3.5k - 5 + 35
⇒ 7(24q) - 2.3.5k + 30
⇒ 7(24q) - 6(5k - 5)
Now as we know that (5k - 5) is a multiple of 4 so in place of that we can write (4p) where (p) belongs to the natural number.
⇒ 7(24q) - 6(4p)
⇒ 24(7q - p)
So as we see this is a multiple of 24.
Thus P (k + 1) is true whenever P (k) is true.
So according to the principle of mathematical induction 2.7k + 3.5k - 5 is divisible by 24.
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