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Mathematics · Class 11 Science

Ch 13Methods of Induction and Binomial Theorem — Class 11 Mathematics, concept-first.

Introduction. The earliest implicit use of a proof by induction is credited to Al-Karaji, around 100 AD; the first explicit statement of the principle as a method is credited to Pascal, in 1665.

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Principle of Mathematical Induction

The Principle of Mathematical Induction (PMI) is a method for proving that a statement , framed for every natural number , is true for all at once — without needing to verify infinitely many individual cases directly.

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4.1

Principle of Mathematical Induction

Introduction. The earliest implicit use of a proof by induction is credited to Al-Karaji, around 100 AD; the first explicit statement of the principle as a method is credited to Pascal, in 1665.

+EXERCISE 4.116 questions
  1. Q1Prove by method of induction, for all $n \in N$: $2 + 4 + 6 + \ldots + 2n = n(n+1)$.Free
  2. Q2Prove by method of induction, for all $n \in N$: $3 + 7 + 11 + \ldots \text{ to } n \text{ terms} = n(2n+1)$.Free
  3. Q3Prove by method of induction, for all $n \in N$: $1^2 + 2^2 + 3^2 + \ldots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$.Free
  4. Q4Prove by method of induction, for all $n \in N$: $1^2 + 3^2 + 5^2 + \ldots + (2n-1)^2 = \dfrac{n}{3}(2n-1)(2n+1)$.Preview
  5. Q5Prove by method of induction, for all $n \in N$: $1^3 + 3^3 + 5^3 + \ldots \text{ to } n \text{ terms} = n^2(2n^2 - 1)$.Preview
  6. Q6Prove by method of induction, for all $n \in N$: $1.2 + 2.3 + 3.4 + \ldots + n(n+1) = \dfrac{n}{3}(n+1)(n+2)$.Preview
  7. Q7Prove by method of induction, for all $n \in N$: $1.3 + 3.5 + 5.7 + \ldots \text{ to } n \text{ terms} = \dfrac{n}{3}(4n^2+6n-1)$.Preview
  8. Q8Prove by method of induction, for all $n \in N$: $\dfrac{1}{1.3} + \dfrac{1}{3.5} + \dfrac{1}{5.7} + \ldots + \dfrac{1}{(2n-1)(2n+1)} = \dfr…Preview
  9. Q9Prove by method of induction, for all $n \in N$: $\dfrac{1}{3.5} + \dfrac{1}{5.7} + \dfrac{1}{7.9} + \ldots \text{ to } n \text{ terms} = \d…Preview
  10. Q10Prove by method of induction, for all $n \in N$: $(2^{3n}-1)$ is divisible by $7$.Preview
  11. Q11Prove by method of induction, for all $n \in N$: $(2^{4n}-1)$ is divisible by $15$.Preview
  12. Q12Prove by method of induction, for all $n \in N$: $3^n - 2n - 1$ is divisible by $4$.Preview
  13. Q13Prove by method of induction, for all $n \in N$: $5 + 5^2 + 5^3 + \ldots + 5^n = \dfrac{5}{4}(5^n - 1)$.Preview
  14. Q14Prove by method of induction, for all $n \in N$: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$.Preview
  15. Q15Given that (recurrence relation) $t_{n+1} = 5t_n + 4$, $t_1 = 4$, prove by induction that (general statement) $t_n = 5^n - 1$.Preview
  16. Q16Prove by method of induction $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^n = \begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}$, $\forall n \i…Preview
4.2

Binomial Theorem for Positive Integral Index

Building up the pattern. Direct multiplication gives: ; ; ; ; . Arranging the coefficients of these five expansions in a triangular array gives Pascal's triangle, and each coefficient can be written a…

+EXERCISE 4.217 questions
  1. Q17Expand $(\sqrt{3}+\sqrt{2})^4$.Free
  2. Q18Expand $(\sqrt{5}-\sqrt{2})^5$.Free
  3. Q19Expand $(2x^2+3)^4$.Free
  4. Q20Expand $\left(2x - \dfrac{1}{x}\right)^6$.Preview
  5. Q21Find the value of $(\sqrt{3}+1)^4 - (\sqrt{3}-1)^4$.Preview
  6. Q22Find the value of $(2+\sqrt{5})^5 + (2-\sqrt{5})^5$.Preview
  7. Q23Prove that $(\sqrt{3}+\sqrt{2})^6 + (\sqrt{3}-\sqrt{2})^6 = 970$.Preview
  8. Q24Prove that $(\sqrt{5}+1)^5 - (\sqrt{5}-1)^5 = 352$.Preview
  9. Q25Using binomial theorem, find the value of $(102)^4$.Preview
  10. Q26Using binomial theorem, find the value of $(1.1)^5$.Preview
  11. Q27Using binomial theorem, find the value of $(9.9)^3$.Preview
  12. Q28Using binomial theorem, find the value of $(0.9)^4$.Preview
  13. Q29Without expanding, find the value of $(x+1)^4 - 4(x+1)^3(x-1) + 6(x+1)^2(x-1)^2 - 4(x+1)(x-1)^3 + (x-1)^4$.Preview
  14. Q30Without expanding, find the value of $(2x-1)^4 + 4(2x-1)^3(3-2x) + 6(2x-1)^2(3-2x)^2 + 4(2x-1)(3-2x)^3 + (3-2x)^4$.Preview
  15. Q31Find the value of $(1.02)^6$, correct upto four places of decimals.Preview
  16. Q32Find the value of $(1.01)^5$, correct upto three places of decimals.Preview
  17. Q33Find the value of $(0.9)^6$, correct upto four places of decimals.Preview
4.3

General Term in the Expansion of (a+b)^n

Naming the terms. In the expansion of , the successive terms are labelled , so that , , , and in general , so that This is called the general term of the expansion: because it is written in terms of r…

+EXERCISE 4.323 questions
  1. Q34In the following expansion, find the indicated term: $\left(2x^2+\dfrac{3}{2x}\right)^8$, 3rd term.Free
  2. Q35In the following expansion, find the indicated term: $\left(x^2-\dfrac{4}{x^3}\right)^{11}$, 5th term.Free
  3. Q36In the following expansion, find the indicated term: $\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^9$, 7th term.Free
  4. Q37In the following expansion, find the indicated term: $\left(\dfrac{1}{3}+a^2\right)^{12}$, 9th term.Preview
  5. Q38In the following expansion, find the indicated term: $\left(3a+\dfrac{4}{a}\right)^{13}$, 10th term.Preview
  6. Q39In the following expansion, find the indicated coefficient: coefficient of $x^3$ in $\left(x^2+\dfrac{3\sqrt{2}}{x}\right)^9$.Preview
  7. Q40In the following expansion, find the indicated coefficient: coefficient of $x^8$ in $\left(2x^5-\dfrac{5}{x^3}\right)^8$.Preview
  8. Q41In the following expansion, find the indicated coefficient: coefficient of $x^9$ in $\left(\dfrac{1}{x}+x^2\right)^{18}$.Preview
  9. Q42In the following expansion, find the indicated coefficient: coefficient of $x^{-3}$ in $\left(x-\dfrac{1}{2x}\right)^5$.Preview
  10. Q43In the following expansion, find the indicated coefficient: coefficient of $x^{-20}$ in $\left(x^3-\dfrac{1}{2x^2}\right)^{15}$.Preview
  11. Q44Find the constant term (term independent of x) in the expansion of $\left(2x+\dfrac{1}{3x^2}\right)^9$.Preview
  12. Q45Find the constant term (term independent of x) in the expansion of $\left(x-\dfrac{2}{x^2}\right)^{15}$.Preview
  13. Q46Find the constant term (term independent of x) in the expansion of $\left(\sqrt{x}-\dfrac{3}{x^2}\right)^{10}$.Preview
  14. Q47Find the constant term (term independent of x) in the expansion of $\left(x^2-\dfrac{1}{x}\right)^9$.Preview
  15. Q48Find the constant term (term independent of x) in the expansion of $\left(2x^2-\dfrac{5}{x}\right)^9$.Preview
  16. Q49Find the middle terms in the expansion of $\left(\dfrac{x}{y}+\dfrac{y}{x}\right)^{12}$.Preview
  17. Q50Find the middle terms in the expansion of $\left(x^2+\dfrac{1}{x}\right)^7$.Preview
  18. Q51Find the middle terms in the expansion of $\left(x^2-\dfrac{2}{x}\right)^8$.Preview
  19. Q52Find the middle terms in the expansion of $\left(\dfrac{x}{a}-\dfrac{a}{x}\right)^{10}$.Preview
  20. Q53Find the middle terms in the expansion of $\left(x^4-\dfrac{1}{x^3}\right)^{11}$.Preview
  21. Q54In the expansion of $(k+x)^8$, the coefficient of $x^5$ is 10 times the coefficient of $x^6$. Find the value of $k$.Preview
  22. Q55Find the term containing $x^6$ in the expansion of $(2-x)(3x+1)^9$.Preview
  23. Q56The coefficient of $x^2$ in the expansion of $(1+2x)^m$ is $112$. Find $m$.Preview
4.4

Middle Term(s) in the Expansion of (a+b)^n

How many terms, and where is the middle? The expansion of always has terms (Remark 1 of Section 4.2). Whether that count is odd or even decides whether there is a single middle term or a pair of them.

+EXERCISE 4.420 questions
  1. Q57State, by writing first four terms, the expansion of $(1+x)^{-4}$, where $|x|<1$.Free
  2. Q58State, by writing first four terms, the expansion of $(1-x)^{1/3}$, where $|x|<1$.Free
  3. Q59State, by writing first four terms, the expansion of $(1-x^2)^{-3}$, where $|x|<1$.Free
  4. Q60State, by writing first four terms, the expansion of $(1+x)^{-1/5}$, where $|x|<1$.Preview
  5. Q61State, by writing first four terms, the expansion of $(1+x^2)^{-1}$, where $|x|<1$.Preview
  6. Q62State, by writing first four terms, the expansion of $(a-b)^{-3}$, where $|b|<|a|$.Preview
  7. Q63State, by writing first four terms, the expansion of $(a+b)^{-4}$, where $|b|<|a|$.Preview
  8. Q64State, by writing first four terms, the expansion of $(a+b)^{1/4}$, where $|b|<|a|$.Preview
  9. Q65State, by writing first four terms, the expansion of $(a-b)^{-1/4}$, where $|b|<|a|$.Preview
  10. Q66State, by writing first four terms, the expansion of $(a+b)^{-1/3}$, where $|b|<|a|$.Preview
  11. Q67Simplify first three terms in the expansion of $(1+2x)^{-4}$.Preview
  12. Q68Simplify first three terms in the expansion of $(1+3x)^{-1/2}$.Preview
  13. Q69Simplify first three terms in the expansion of $(2-3x)^{1/3}$.Preview
  14. Q70Simplify first three terms in the expansion of $(5+4x)^{-1/2}$.Preview
  15. Q71Simplify first three terms in the expansion of $(5-3x)^{-1/3}$.Preview
  16. Q72Use binomial theorem to evaluate the following upto four places of decimals: $\sqrt{99}$.Preview
  17. Q73Use binomial theorem to evaluate the following upto four places of decimals: $\sqrt[3]{126}$.Preview
  18. Q74Use binomial theorem to evaluate the following upto four places of decimals: $\sqrt[4]{16.08}$.Preview
  19. Q75Use binomial theorem to evaluate the following upto four places of decimals: $(1.02)^{-5}$.Preview
  20. Q76Use binomial theorem to evaluate the following upto four places of decimals: $(0.98)^{-3}$.Preview
4.5

Binomial Theorem for Negative Index or Fraction

Why a new form is needed. The Binomial Theorem of Section 4.2 was stated using , which requires to be a non-negative integer so that is defined.

+EXERCISE 4.57 questions
  1. Q77Show that $C_0 + C_1 + C_2 + \ldots + C_8 = 256$.Free
  2. Q78Show that $C_0 + C_1 + C_2 + \ldots + C_9 = 512$.Free
  3. Q79Show that $C_1 + C_2 + C_3 + \ldots + C_7 = 127$.Free
  4. Q80Show that $C_1 + C_2 + C_3 + \ldots + C_6 = 63$.Preview
  5. Q81Show that $C_0+C_2+C_4+C_6+C_8=C_1 +C_3+C_5+C_7 = 128$.Preview
  6. Q82Show that $C_1 + C_2+ C_3 + \ldots + C_n = 2^n - 1$.Preview
  7. Q83Show that $C_0+2C_1+3C_2+4C_3+\ldots+(n+1)C_n=(n+2)2^{n-1}$.Preview
4.6

Binomial Coefficients

Naming the coefficients. The coefficients occurring in the expansion of are called the binomial coefficients, and for brevity are written .

More questions

46 Q
+Show 10 questions10 questions
  1. Q84Select the correct answer from the given alternatives. The total number of terms in the expression of $(x+y)^{100} + (x-y)^{100}$ after simp…Free
  2. Q85Select the correct answer from the given alternatives. The middle term in the expansion of $(1+x)^{2n}$ will be: (A) $(n-1)$th (B) $n$th (C)…Free
  3. Q86Select the correct answer from the given alternatives. In the expansion of $(x^2-2x)^{10}$, the coefficient of $x^{16}$ is: (A) $-1680$ (B)…Free
  4. Q87Select the correct answer from the given alternatives. The term not containing $x$ in the expansion of $(1-x)^2\left(x+\dfrac{1}{x}\right)^{…Preview
  5. Q88Select the correct answer from the given alternatives. The number of terms in the expansion of $(4y+x)^8-(4y-x)^8$ is: (A) 4 (B) 5 (C) 8 (D)…Preview
  6. Q89Select the correct answer from the given alternatives. The value $^{14}C_1 + {}^{14}C_3+ {}^{14}C_5+ \ldots + {}^{14}C_{11}$ is: (A) $2^{14}…Preview
  7. Q90Select the correct answer from the given alternatives. The value $^{11}C_2 + {}^{11}C_4+ {}^{11}C_6+ {}^{11}C_8$ is equal to: (A) $2^{10}-1$…Preview
  8. Q91Select the correct answer from the given alternatives. In the expansion of $(3x+2)^4$, the coefficient of middle term is: (A) 36 (B) 54 (C)…Preview
  9. Q92Select the correct answer from the given alternatives. The coefficient of the 8th term in the expansion of $(1+x)^{10}$ is: (A) 7 (B) 120 (C…Preview
  10. Q93Select the correct answer from the given alternatives. If the coefficient of $x^2$ and $x^3$ in the expansion of $(3+ax)^9$ are the same, th…Preview
+Show 36 questions36 questions
  1. Q94Prove, by method of induction, for all $n \in N$: $8 + 17 + 26 + \ldots + (9n-1) = \dfrac{n}{2}(9n+7)$.Free
  2. Q95Prove, by method of induction, for all $n \in N$: $1^2 + 4^2 + 7^2 + \ldots + (3n-2)^2 = \dfrac{n}{2}(6n^2-3n -1)$.Free
  3. Q96Prove, by method of induction, for all $n \in N$: $2 + 3.2 + 4.2^2 + \ldots + (n+1)2^{n-1} = n. 2^n$.Free
  4. Q97Prove, by method of induction, for all $n \in N$: $\dfrac{1}{3.4.5} + \dfrac{2}{4.5.6} + \dfrac{3}{5.6.7} + \ldots + \dfrac{n}{(n+2)(n+3)(n+…Preview
  5. Q98Given that $t_{n+1} = 5t_n - 8$, $t_1 = 3$, prove by method of induction that $t_n = 5^{n-1} +2$.Preview
  6. Q99Prove by method of induction $\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & -4n \\ n & -2n+1 \end{pmatrix}$, $\f…Preview
  7. Q100Expand $(3x^2 + 2y)^5$.Preview
  8. Q101Expand $\left(\dfrac{2x}{3}-\dfrac{3}{2x}\right)^4$.Preview
  9. Q102Find third term in the expansion of $\left(9x^2-\dfrac{y^3}{6}\right)^4$.Preview
  10. Q103Find tenth term in the expansion of $\left(2x^2+\dfrac{1}{x}\right)^{12}$.Preview
  11. Q104Find the middle term (s) in the expansion of $\left(\dfrac{2a}{3}-\dfrac{3}{2a}\right)^6$.Preview
  12. Q105Find the middle term (s) in the expansion of $\left(x-\dfrac{1}{2y}\right)^{10}$.Preview
  13. Q106Find the middle term (s) in the expansion of $(x^2+2y^2)^7$.Preview
  14. Q107Find the middle term (s) in the expansion of $\left(\dfrac{3x^2}{2}-\dfrac{1}{3x}\right)^9$.Preview
  15. Q108Find the coefficient of $x^6$ in the expansion of $\left(3x^2-\dfrac{1}{3x}\right)^9$.Preview
  16. Q109Find the coefficient of $x^{60}$ in the expansion of $\left(\dfrac{1}{x^2}+x^4\right)^{18}$.Preview
  17. Q110Find the constant term in the expansion of $\left(\dfrac{4x^2}{3}+\dfrac{3}{2x}\right)^9$.Preview
  18. Q111Find the constant term in the expansion of $\left(2x^2-\dfrac{1}{x}\right)^{12}$.Preview
  19. Q112Prove the following by using method of induction: $\log_a x^n = n \log_a x$, $x > 0$, $n \in N$.Preview
  20. Q113Prove the following by using method of induction: $15^{2n-1}+1$ is divisible by $16$, for all $n \in N$.Preview
  21. Q114Prove the following by using method of induction: $5^{2n} - 2^{2n}$ is divisible by $3$, for all $n \in N$.Preview
  22. Q115If the coefficient of $x^{16}$ in the expansion of $(x^2 + ax)^{10}$ is $3360$, find $a$.Preview
  23. Q116If the middle term in the expansion of $\left(x+\dfrac{b}{x}\right)^6$ is $160$, find $b$.Preview
  24. Q117If the coefficient of $x^2$ and $x^3$ in the expansion of $(3 + kx)^9$ are equal, find $k$.Preview
  25. Q118If the constant term in the expansion of $\left(x^3+\dfrac{k}{x^8}\right)^{11}$ is $1320$, find $k$.Preview
  26. Q119Show that there is no term containing $x^6$ in the expansion of $\left(x^2-\dfrac{3}{x}\right)^{11}$.Preview
  27. Q120Show that there is no constant term in the expansion of $\left(2x-\dfrac{x^2}{4}\right)^9$.Preview
  28. Q121State, first four terms in the expansion of $\left(1-\dfrac{2x}{3}\right)^{-1/2}$.Preview
  29. Q122State, first four terms in the expansion of $(1-x)^{-1/4}$.Preview
  30. Q123State, first three terms in the expansion of $(5 + 4x)^{-1/2}$.Preview
  31. Q124Using binomial theorem, find the value of $\sqrt[3]{995}$ upto four places of decimals.Preview
  32. Q125Find approximate value of $\dfrac{1}{\sqrt[4]{4.08}}$ upto four places of decimals.Preview
  33. Q126Find the term independent of x in the expansion of $(1 -x^2)\left(x+\dfrac{2}{x}\right)^6$.Preview
  34. Q127$(a + bx) (1 - x)^6 = 3 -20x + cx^2 + \ldots$, then find $a$, $b$, $c$.Preview
  35. Q128The 3rd term of $(1+x)^n$ is $36x^2$. Find 5th term.Preview
  36. Q129Suppose $(1+kx)^n = 1-12x + 60x^2 - \ldots$, find $k$ and $n$.Preview