4. 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) =<math><mrow><mfrac><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math>

4. Let the given statement be P(n) i.e.,P(n): 1.2.3 + 2.3.4 + … + n (n + 1)(n + 2) = n(n+1)(n+2)(n+3)4If n=1, we getP(1): 1.2.3 = 6 = 1(1+1)(1+2)(1+3)4 = 1.2.3.44=6which is true.considerP(k) is true for some positive integer k1.2.3 + 2.3.4 + … + k(k + 1)(k + 2) = k(k+1)(k+2)(k+3)4 -------------------(1)Now, let us prove that P(k+1) is true.Here,1.2.3 + 2.3.4 + … + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3)By eqn (1), we get,= k(k+1)(k+2)(k+3)4+(k+1)(k+2)(k+3)=(k+1)(k+2)(k+3) [k4+1]= (k+1)(k+2)(k+3)(k+4)4By further Simplification,(k+1)(k+1+1)(k+1+2)(k+1+3)4? P(k+1)is true whenever P(k) is true.Hence, from the principle of mathematical induction, the P(n) is true for all natural numbers n.

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Ncert Solutions Maths class 11th Maths Class 11th Principle of Mathematical Induction

4. 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) = $\frac{n (n + 1) (n + 2) (n + 3)}{4}$

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Payal Gupta | Contributor-Level 10

4 months ago

4. Let the given statement be P(n) i.e.,

P(n): 1.2.3 + 2.3.4 + … + n (n + 1)(n + 2) = $\frac{n (n + 1) (n + 2) (n + 3)}{4}$

If n=1, we get

P(1): 1.2.3 = 6 = $\frac{1 (1 + 1) (1 + 2) (1 + 3)}{4}$ = $\frac{1.2.3.4}{4} = 6$

which is true.

considerP(k) is true for some positive integer k

1.2.3 + 2.3.4 + … + k(k + 1)(k + 2) = $\frac{k (k + 1) (k + 2) (k + 3)}{4}$ -------------------(1)

Now, let us prove that P(k+1) is true.

Here,1.2.3 + 2.3.4 + … + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3)

By eqn (1), we get,

= $\frac{k (k + 1) (k + 2) (k + 3)}{4} + (k + 1) (k + 2) (k + 3)$

=(k+1)(k+2)(k+3) $[\frac{k}{4} + 1]$

= $\frac{(k + 1) (k + 2) (k + 3) (k + 4)}{4}$

By further Simplification,

$\frac{(k + 1) (k + 1 + 1) (k + 1 + 2) (k + 1 + 3)}{4}$

? P(k+1)is true whenever P(k) is true.

Hence, from the principle of mathematical induction, the P(n) is true for all natural numbers n.

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Payal Gupta

24. Let P (n) be the statement “ 2n+7< (n+3)2”

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9<16 which is true. This P (1) is true.

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$= {(k + 4)}^{2} - (2 k - 5)$

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$\Rightarrow P (k + 1)$ is true.

$∴$ By the principle of mathematical induction, P (n)is true for all n $\in$ N.

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