Algebra – Mathematical Induction Quiz - ICSE Class 11 Mathematics

Free practice quiz: Algebra – Mathematical Induction (ICSE Class 11 Mathematics). 57 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 57 multiple-choice questions on Algebra – Mathematical Induction, and 57 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. Let $P(n)$ be the statement: $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n(n+1) = \frac{n(n+1)(n+2)}{3}$ for all positive integers $n$. If $P(k)$ is true, what is $P(k+1)$?

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

   Answer: A. $\frac{k(k+1)(k+2)}{3} + (k+1)(k+2) = \frac{(k+1)(k+2)(k+3)}{3}$

   To prove $P(k+1)$, we assume $P(k)$ is true and show that $P(k+1)$ is also true. $P(k)$ states that the sum up to $k$ terms is $\frac{k(k+1)(k+2)}{3}$. The sum for $P(k+1)$ includes the $(k+1)$-th term, which is $(k+1)((k+1)+1) = (k+1)(k+2)$. Therefore, $P(k+1)$ is the sum of $P(k)$ and the $(k+1)$-th term: $\frac{k(k+1)(k+2)}{3} + (k+1)(k+2)$. We then need to show this equals the formula for $n=k+1$, which is $\frac{(k+1)((k+1)+1)((k+1)+2)}{3} = \frac{(k+1)(k+2)(k+3)}{3}$. The correct option represents this step.

2. Let $P(n)$ be the statement: $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots + n(n+1)(n+2) = \frac{n(n+1)(n+2)(n+3)}{4}$. If $P(k)$ is true, what is the expression for $P(k+1)$?

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

   Answer: A. $\frac{k(k+1)(k+2)(k+3)}{4} + (k+1)(k+2)(k+3) = \frac{(k+1)(k+2)(k+3)(k+4)}{4}$

   Assuming $P(k)$ is true, the sum of the first $k$ terms is $\frac{k(k+1)(k+2)(k+3)}{4}$. The $(k+1)$-th term in the series is $(k+1)((k+1)+1)((k+1)+2) = (k+1)(k+2)(k+3)$. To get $P(k+1)$, we add this $(k+1)$-th term to the sum of the first $k$ terms: $\frac{k(k+1)(k+2)(k+3)}{4} + (k+1)(k+2)(k+3)$. We then need to show this equals the formula for $n=k+1$, which is $\frac{(k+1)((k+1)+1)((k+1)+2)((k+1)+3)}{4} = \frac{(k+1)(k+2)(k+3)(k+4)}{4}$. Option 1 correctly represents this inductive step.

3. Consider the statement: $1^2 - 2^2 + 3^2 - 4^2 + \dots + (-1)^{n-1}n^2 = (-1)^{n-1} \frac{n(n+1)}{2}$. What is the base case for this statement?

   1. n=1
   2. n=2
   3. n=0
   4. n=-1
   Show answer

   Answer: A. n=1

   The statement is for all positive integers $n$. The smallest positive integer is $n=1$. For $n=1$, the left side is $(-1)^{1-1}1^2 = (-1)^0 \cdot 1 = 1$. The right side is $(-1)^{1-1} \frac{1(1+1)}{2} = (-1)^0 \frac{1 \cdot 2}{2} = 1 \cdot 1 = 1$. Since $1=1$, the statement is true for $n=1$.

4. Let $P(n)$ be the statement: $\sum_{i=1}^n \frac{1}{i(i+1)} = \frac{n}{n+1}$. What is the $(k+1)$-th term in the summation?

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

   Answer: B. $\frac{1}{(k+1)(k+2)}$

   The summation is given by $\sum_{i=1}^n \frac{1}{i(i+1)}$. The $(k+1)$-th term corresponds to the value of the expression when $i = k+1$. Substituting $i = k+1$ into the general term $\frac{1}{i(i+1)}$, we get $\frac{1}{(k+1)((k+1)+1)} = \frac{1}{(k+1)(k+2)}$.

5. Which of the following is a typical domain for mathematical induction proofs?

   1. All real numbers.
   2. All positive integers.
   3. All rational numbers.
   4. All complex numbers.
   Show answer

   Answer: B. All positive integers.

   Mathematical induction is primarily used to prove statements that hold for all positive integers (or sometimes for all non-negative integers, or for all integers greater than or equal to some specific integer). Its recursive nature relies on the well-ordered property of these sets.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Using the principle of mathematical induction, prove that for all positive integers $n$, $3^{2n} - 1$ is divisible by 8.
2. Which of the following statements is a necessary condition for proving a statement $P(n)$ by mathematical induction for all positive integers $n$?
3. Let $P(n)$ be the statement: $a + (a+d) + (a+2d) + \dots + (a+(n-1)d) = \frac{n}{2}[2a + (n-1)d]$. If $P(k)$ is true, then $P(k+1)$ is obtained by:
4. Consider the statement $P(n)$: $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$. If $P(k)$ is true, what is the expression for $P(k+1)$?
5. For what value of $n$ is the statement $P(n): n > 5$ first proven true by mathematical induction?
6. Let $P(n)$ be the statement: $\sum_{i=1}^n i = \frac{n(n+1)}{2}$. If $P(k)$ is true, what is the expression for the sum of the first $k+1$ natural numbers?
7. If $P(k)$ implies $P(k+1)$ for all $k \ge m$, and $P(m)$ is true, what can be concluded about $P(n)$ for all $n \ge m$?
8. What is the base case in the induction proof for the statement: $2+4+6+\dots+2n = n(n+1)$ for all positive integers $n$?
9. Let $P(n)$ be the statement: $1 imes 3 + 2 imes 4 + 3 imes 5 + \dots + n(n+2) = \frac{n(n+1)(n+5)}{6}$. If $P(k)$ is true, what is the expression for $P(k+1)$?
10. What is the minimum value of $n$ for which the statement '$n! > 2^n$' is proven true by mathematical induction?
11. Which of the following is NOT a part of the mathematical induction proof for $P(n)$ for all $n \ge 1$?
12. Let $P(n)$ be the statement: $1^3 + 2^3 + \dots + n^3 = \left(\frac{n(n+1)}{2} ight)^2$. If $P(k)$ is true, what is the expression for $P(k+1)$?
13. What is the inductive hypothesis in the proof that $1+2+3+\dots+n = \frac{n(n+1)}{2}$?
14. Let $P(n)$ be the statement: $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n(n+1)} = \frac{n}{n+1}$. If $P(k)$ is true, what is the expression for $P(k+1)$?
15. Which of the following is a valid inductive step for proving $P(n)$ is true for all $n \ge m$?
16. Let $P(n)$ be the statement: $1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \dots + n \cdot n! = (n+1)! - 1$. If $P(k)$ is true, what is the expression for $P(k+1)$?
17. If the base case of a proof by induction is for $n=0$, and the inductive step shows $P(k) \implies P(k+1)$, for what values of $n$ is $P(n)$ true?
18. Consider the statement: $a^n - b^n$ is divisible by $a-b$ for all positive integers $n$. What is the base case that needs to be verified?
19. In a proof by induction, if the base case is true and the inductive step is valid, what does this guarantee?
20. Let $P(n)$ be the statement: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n} = 1 - \frac{1}{2^n}$. If $P(k)$ is true, what is the expression for $P(k+1)$?
21. Let $P(n)$ be the statement: $n^2 + n$ is divisible by 2 for all positive integers $n$. Which of the following is the correct form of the inductive step?
22. If $P(k)$ is true, and $P(k) \implies P(k+1)$ is proven, for a statement $P(n)$ valid for $n \ge 1$, what is the smallest value of $n$ for which $P(n)$ is guaranteed to be true?
23. Let $P(n)$ be the statement: $3^n > n^2$ for all positive integers $n$. If $P(k)$ is true, what must be shown in the inductive step?
24. Which of the following is an example of a statement that can be proven using mathematical induction?
25. If $P(n)$ is a statement about natural numbers, and we prove $P(1)$ and $P(k) \implies P(k+1)$, what can we conclude?
26. Let $P(n)$ be the statement: $\sum_{i=1}^n (2i-1) = n^2$. What is the $(k+1)$-th term in the sum?
27. If $P(k)$ is true and $P(k) \implies P(k+1)$, this means that the statement holds for:
28. Let $P(n)$ be the statement: $2+4+6+...+2n = n(n+1)$. If $P(k)$ is true, then $P(k+1)$ is proven by showing:
29. Let $P(n)$ be the statement: $1 + 2 + \dots + n = \frac{n(n+1)}{2}$. What is the base case verification for $n=1$?
30. In the inductive step, if we assume $P(k)$ is true, what is the goal for $P(k+1)$?
31. Let $P(n)$ be the statement: $a^n - b^n$ is divisible by $a-b$. If $P(k)$ is true, which of the following is the correct form for the inductive step?
32. If a statement $P(n)$ is proven by induction for all integers $n \ge m$, and we verify $P(m)$ and $P(k) \implies P(k+1)$, what can we say about $P(m+1)$?
33. Consider the statement: $1 \cdot 2 + 2 \cdot 3 + \dots + n(n+1) = \frac{n(n+1)(n+2)}{3}$. Which of the following is the correct form of the inductive step, assuming $P(k)$ is true?
34. If a statement $P(n)$ is to be proven for all integers $n \ge 5$, what is the base case?
35. Let $P(n)$ be the statement: $n < 2^n$ for all positive integers $n$. If $P(k)$ is true, what must be shown in the inductive step?
36. Which of the following correctly describes the conclusion of a valid induction proof?
37. Let $P(n)$ be the statement: $1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{n(2n-1)(2n+1)}{3}$. If $P(k)$ is true, what is the $(k+1)$-th term in the sum?
38. Which of the following is not a component of a standard proof by mathematical induction?
39. Let $P(n)$ be the statement: $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$. If $P(k)$ is true, what is the expression that needs to be simplified to prove $P(k+1)$?
40. What is the significance of the base case in a proof by mathematical induction?
41. Let $P(n)$ be the statement: $n \ge 2$ implies $2^n > n+1$. If $P(k)$ is true, what must be shown in the inductive step?
42. Let $P(n)$ be the statement: $1 \cdot 2 + 2 \cdot 3 + \dots + n(n+1) = \frac{n(n+1)(n+2)}{3}$. What is the base case verification for $n=1$?
43. If $P(k)$ implies $P(k+1)$ for all $k \ge m$, and $P(m)$ is true, what is the smallest integer $n$ for which $P(n)$ is guaranteed to be true?
44. Let $P(n)$ be the statement: $\sum_{i=1}^n i = \frac{n(n+1)}{2}$. If $P(k)$ is true, what is the expression for the sum of the first $k+1$ natural numbers in terms of $P(k)$?
45. Consider the statement: $1 imes 1! + 2 imes 2! + \dots + n imes n! = (n+1)! - 1$. What is the $(k+1)$-th term in the sum?
46. If a statement $P(n)$ is true for $n=1$ and $P(k)$ true implies $P(k+1)$ true, then $P(n)$ is true for all integers $n \ge 1$. This is the principle of:
47. Let $P(n)$ be the statement: $3^{2n} - 1$ is divisible by 8. If $P(k)$ is true, what is the form of the inductive step?
48. What is the smallest integer $n$ for which the statement '$n > \log_2(n)$' is true?
49. Let $P(n)$ be the statement: $1 \cdot 2 + 2 \cdot 3 + \dots + n(n+1) = \frac{n(n+1)(n+2)}{3}$. If $P(k)$ is true, what is the inductive hypothesis?
50. Which of the following statements cannot be proven by mathematical induction?
51. Let $P(n)$ be the statement: $\sum_{i=0}^n x^i = \frac{x^{n+1}-1}{x-1}$ for $x e 1$. If $P(k)$ is true, what is the correct expression for $P(k+1)$?
52. If $P(n)$ is a statement, and the inductive step proves $P(k) \implies P(k+1)$, this means that:

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