PRINCIPLE OF MATHEMATICAL INDUCTION - Gujarat Board (GSEB) STD 11 Science MATHS Questions

208

Questions

2

Question groups

5

Question types

01

MCQ

185 Q→02

(Each question 4 marks)

23 Q→

Sample Questions

PRINCIPLE OF MATHEMATICAL INDUCTION questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1MCQ1 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 2MCQ1 Mark

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

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

Answer: C.

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Q 3MCQ1 Mark

Let P(n) be a statement and P(n)=P(n+1)∀n ∈ 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 4MCQ1 Mark

If P(n) = 2 + 4 + ......+ 2n, n ϵ N, then P(k) = k(k + 1) + 2 ⇒ P(k) = k(k + 1) + 2 for all k ϵ N. S we can conclude that P(n) = n(n + 1) + 2 for

A
all n ϵ N
B
n > 1
C
n > 2
✓
nothing can be said

Answer: D.

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Q 5MCQ1 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 6(Each question 4 marks)4 Marks

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

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Q 7(Each question 4 marks)4 Marks

Prove the following by using the principle of mathematical induction for all n ∈ N:$1^2+3^2+5^2+...+(2\text{n}-1)^2=\frac{\text{n}(2\text{n}-1)(2\text{n}+1)}{3}.$

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Q 8(Each question 4 marks)4 Marks

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

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Q 9(Each question 4 marks)4 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 10(Each question 4 marks)4 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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