Mathematical Induction - Rajasthan Board (RBSE) STD 11 Science MATHS Questions

If p(n): 2n < (1 × 2 × 3 × ... × n). Then the smallest positive integer for which p(n) is true is:

If $10^\text{n} + 3 \times 4^{\text{n}+2}+\lambda$ is divisible by 9 for all $\text{n}\in\text{N},$ then the least positive integer value of $\lambda$ is

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Q 3M.C.Q (1 Marks)1 Mark

If $\text{i}^2=-1,$ then the sum $\text{i}+\text{i}^2+\text{i}^3+...$ upto 1000 terms is equal to:

1
-1
i
0

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Q 4M.C.Q (1 Marks)1 Mark

A student was asked to prove a statement p(n) by induction. He proved p(K + 1) is true whenever p(k) is true for all $\text{k}>5\in\text{N}$ and also p(5) is true. On the basis of this he could conclude that p(n) is true.

For all $\text{n}\in\text{N}$
For all n > 5
For all $\text{n}\geq5$
For all n > 5

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Q 5M.C.Q (1 Marks)1 Mark

If xⁿ - 1 is divisible by $\text{x}-\lambda,$ then the least positive integral value of $\lambda$ is:

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Q 61 Marks Question1 Mark

State the first principle of mathematical induction.

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Q 71 Marks Question1 Mark

Write the set of value if n for which the statement p(n): 2n < n! is true.

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Q 81 Marks Question1 Mark

State the second principle of mathematical induction.

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Q 91 Marks Question1 Mark

If p(n): 2 × 4^2n-1 + 3³ⁿ⁺¹ is divisible by $\lambda$ for all $\text{n}\in\text{N}$ is true, then find the value of $\lambda$

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Q 101 Marks Question1 Mark

If P(n) is the statement "n(n + 1) is even", then what is P(3)?

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Q 112 Marks Questions2 Marks

If P(n) is the statement "n³ + n is divisible by 3", prove that P(3) is true but P(4) is not true.

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Q 123 Marks Question3 Marks

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

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Q 133 Marks Question3 Marks

If P(n) is the statement "n² - n + 41 is prime", prove that P(1), P(2) and P(3) are true. Prove also that P(41) is not true.

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Q 143 Marks Question3 Marks

If P(n) is the statement "2ⁿ ≥ 3n" and if P(r) is true, prove that P(r + 1) is true.

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Q 155 Marks Questions5 Marks

Prove that $\cos\alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+...+\cos(\alpha+(\text{n}-1)\beta)\\=\frac{\cos\Big\{\alpha+\big(\frac{\text{n}-1}{2}\big)\beta\Big\}\sin\big(\frac{\text{n}\beta}{2}\big)}{\sin\frac{\beta}{2}}$ For all $\text{n}\in\text{N}.$

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Q 165 Marks Questions5 Marks

A sequence x₁, x₂, x₃, ... is defined by letting x₁ = 2 and $\text{x}_{\text{k}}=\frac{\text{x}_{\text{k}}-1}{\text{n}}$ for all natural numbers k, $\text{k}\geq2.$ Show that $\text{x}_{\text{n}}=\frac{2}{\text{n}!}$ for all$\text{n}\in\text{N}.$

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Q 175 Marks Questions5 Marks

Prove the following by the principle of mathematical induction:
$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+...+\frac{1}{(\text{3n-1)(3n+2)}}=\frac{\text{n}}{\text{6n}+4}$

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Q 185 Marks Questions5 Marks

Prove the following by the principle of mathematical induction:
$\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+...+\frac{1}{(\text{4n-1)(4n+3)}}=\frac{\text{n}}{3(\text{4n}+3)}$

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Q 195 Marks Questions5 Marks

Prove the following by the principle of mathematical induction:
$1 + 3 + 3^2 + ... + 3^{\text{n}-1}=\frac{3^\text{n}-1}{2}$

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Mathematical Induction question types

Mathematical Induction questions

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Mathematical Induction MATHS - Mathematical Induction

Mathematical Induction question types

Mathematical Induction questions

Generate a Mathematical Induction paper free