Prove that 1n+1+1n+2+...+12n>1324, for all natural numbers n > 1.

Question

Prove that `1/(n + 1) + 1/(n + 2) + ... + 1/(2n) > 13/24`, for all natural numbers n > 1.

Theorem

Solution

Let P(n): `1/(n + 1) + 1/(n + 2) + ... + 1/(2n) > 13/24` ∀ n ∈ N

Step 1: P(2) : `1/(2 + 1) + 1/(2 + 2) > 13/24`

⇒ `1/3 + 1/4 > 13/24`

⇒ `7/12 > 13/24`

⇒ `14/24 > 13/24` which is true for P(2).

Step 2: P(k) : `1/(k + 1) + 1/(k + 2) + ... + 1/(2k) > 13/24`.

Let it be true for P(k).

Step 3: P(k + 1) : `1/(k + 1) + 1/(k + 2) + ... + 1/(2k) + 1/(2(k + 1)) > 13/24`

Since `1/(k + 1) + 1/(k + 2) + ... + 1/(2k) > 13/24`

So `1/(k + 1) + 1/(k + 2) + ... + 1/(2k) + 1/(2(k + 1)) > 13/24`

Which is true for P(k + 1).

Hence, P(k + 1) is true whenever P(k) is true.

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Principle of Mathematical Induction

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Q 24 Q 23 Q 25

Chapter 4: Principle of Mathematical Induction - Exercise [Page 72]

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NCERT Exemplar Mathematics Exemplar [English] Class 11

Chapter 4 Principle of Mathematical Induction
Exercise | Q 24 | Page 72

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