Principle of Mathematical Induction - CBSE STD 11 Science Maths Questions

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

If $x^{2 n-1}+y^{2 n-1}$ is divisible by $x + y,$ if $n$ is:

✓
a positive integer
B
an even positive integer
C
an odd positive integer
D
None of these

Answer: A.

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Q 2M.C.Q (1 Marks)1 Mark

The nth terms of the series $3 + 7 + 13 + 21 +……….$ is

A
$4n – 1$
B
$2n + 1$
✓
$n^2 + n + 1$
D
$n + 2$

Answer: C.

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

Let $P(n)$ be a statement and $P(n)=P(n+1)\forall n \in N,$ then $P(n)$ is true for what values of $n?$

✓
For all $n$
B
For all $n>1$
C
For all $n>m , m$ being a fixed positive integer
D
Nothing can be said

Answer: A.

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

If $P(n) = 2 + 4 + ......+ 2n, n\ \epsilon\ N,$ then $P(k) = k(k + 1) + 2 \Rightarrow P(k) = k(k + 1) + 2$ for all $k\ \epsilon\ N. S$ we can conclude that $P(n) = n(n + 1) + 2$ for

A
all $n\ \epsilon\ N$
B
$n > 1$
C
$n > 2$
✓
nothing can be said

Answer: D.

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

For natural number $n, 2^n, (n – 1) ! < ,$ if:

A
$n < 2$
✓
$n > 2$
C
$n ≥ 2$
D
never

Answer: B.

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Q 6True False[1 Marks ]1 Mark

State whether the following statement is true or false. Justify:
Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.

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Q 71 Marks Question1 Mark

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

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Q 81 Marks Question1 Mark

State the first principle of mathematical induction.

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Q 91 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 101 Marks Question1 Mark

If $P(n)$ is a statement $(n \in N )$ such that, if $P(k)$ is true, $P(k +1)$ is true for $k \in N ,$ then $P(n)$ is true:

A
for all $n$
B
for all $n > 1$
✓
for all $n > 2$
D
Nothing can be said

Answer: C.

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Q 111 Marks Question1 Mark

State the second principle of mathematical induction.

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Q 122 Marks Questions2 Marks

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

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Q 133 Marks Question3 Marks

Prove the following by using the principle of mathematical induction for all n ∈ N:
$\Big(1+\frac{1}{1}\Big)\Big(1+\frac{1}{2}\Big)\Big(1+\frac{1}{3}\Big)...\Big(1+\frac{1}{\text{n}}\Big)=(\text{n+1}).$

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Q 143 Marks Question3 Marks

If P(n) is the statement "$n^2 + n$ is even", and if $P(r)$ is true, then $P(r + 1)$ is true.

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Q 153 Marks Question3 Marks

Give an example of a statement P(n) which is true for all $\text{n}\geq4$ but P(1), P(2) and P(3) are not true. Justify your answer.

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Q 163 Marks Question3 Marks

If $P(n)$ is the statement " $n^2-n+41$ 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 173 Marks Question3 Marks

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

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Q 18Fill In The Blanks[1 Marks ]1 Mark

Fill in the blanks:
If P(n)P: 2n < n!, n ∈ N, then P(n) is true for all $\text{n}\geq$ ________.

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Q 195 Marks Questions5 Marks

Prove the following by using the principle of mathematical induction for all $n \in N:$
$n(n + 1)(n + 5)$ is a multiple of $3.$

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Q 205 Marks Questions5 Marks

Prove the following by using the principle of mathematical induction for all $n \in N:$
$41^n – 14^n $ is a multiple of $27.$

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Q 215 Marks Questions5 Marks

Prove the following by using the principle of mathematical induction for all n ∈ N:$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2^\text{n}}=1-\frac{1}{2^{\text{n}}}.$

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Q 225 Marks Questions5 Marks

Prove the following by using the principle of mathematical induction for all n ∈ N:
$1+2+3+...+\text{n}<\frac{1}{8}(2\text{n}+1)^2.$

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Q 235 Marks Questions5 Marks

Prove the following by using the principle of mathematical induction for all n ∈ N:$1+3+3^2+....+3^{\text{n}-1}=\frac{(3^{\text{n}}-1)}{2}.$

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Principle of Mathematical Induction questions

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Principle of Mathematical Induction Maths - Principle of Mathematical Induction

Principle of Mathematical Induction question types

Principle of Mathematical Induction questions

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