If P (N) is the Statement "N2 + N is Even", and If P (R) is True, Then P (R + 1) is True.

Question

If P (n) is the statement "n² + n is even", and if P (r) is true, then P (r + 1) is true.

Solution

\[P(n): n^2 + n is even . \]
\[Also, \]
\[P(r) is true . \]
\[Thus, r^2 + r is even . \]
\[To prove: P(r + 1) is true . \]
\[Now, \]
\[P(r + 1) = (r + 1 )^2 + r + 1\]
\[ = r^2 + 1 + 2r + r + 1 \]
\[ = r^2 + 3r + 2\]
\[ = r^2 + r + 2r + 2\]
\[ = P(r) + 2(r + 1)\]
\[P(r) \text{ is even } . \]
\[Also, 2(r + 1)\text{ is even, as it is a multiple of 2 } . \]
\[\text{ Therefore, P(r + 1) is even and true } . \]

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Principle of Mathematical Induction

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Q 4 Q 3 Q 5

Chapter 12: Mathematical Induction - Exercise 12.1 [Page 3]

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R.D. Sharma Mathematics [English] Class 11

Chapter 12 Mathematical Induction
Exercise 12.1 | Q 4 | Page 3

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