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Mathematics · Class 11 Science

Ch 9Binomial Theorem — Class 11 Mathematics, concept-first.

Long before the algebraic notation existed, mathematicians in several ancient civilisations had already worked out the pattern of coefficients that appears when a binomial such as is raised to a power.

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Key concepts

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General Term in Binomial Expansion

In the expansion of , the individual terms are labelled in order, and the single formula that produces any one of them directly is the general term The subscript convention takes a moment to internalise: the term numbere…

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Chapter contents

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1

A Brief History of the Binomial Theorem

Long before the algebraic notation existed, mathematicians in several ancient civilisations had already worked out the pattern of coefficients that appears when a binomial such as is raised to a power…

2

Statement of the Binomial Theorem for Positive Integral Index

We already know how to expand small positive-integer powers of a binomial by repeated multiplication:

3

Proof by Mathematical Induction

Let be the statement We prove is true for every positive integer using the principle of mathematical induction.

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Pascal's Triangle

Section 3 proved the key identity : each binomial coefficient of index is the sum of two neighbouring coefficients of index .

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The General Term in the Expansion

Writing out the binomial theorem in full, notice that the first term () is the st term, the second term () is the nd term, and so on -- the term containing is always the th term of the expansion, beca…

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The Middle Term(s)

The expansion of always has terms (Section 2, Pattern (i)). Whether this total, , is odd or even depends entirely on whether itself is even or odd, and that parity decides whether the expansion has a…

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Simple Applications

When is small (say , and often much smaller), the terms of shrink rapidly, because each successive term carries one more power of the small number .

Summary

This chapter developed the binomial theorem for a positive integral index from first principles. Starting from a brief history of the coefficient pattern -- known in ancient India as Meru-prastara and…

More questions

28 Q
+Show 3 questions3 questions
  1. Q26If the coefficients of the $2$nd, $3$rd and $4$th terms in the expansion of $(1+x)^n$ are in arithmetic progression, find the value of $n$.Free
  2. Q27Find the term independent of $x$ in the expansion of $\left(\dfrac{3}{2}x^2-\dfrac{1}{3x}\right)^9$.Preview
  3. Q28Using the factorial formula $^{n}C_r = \dfrac{n!}{r!(n-r)!}$, prove algebraically that $^{n}C_{r-1} + {}^{n}C_r = {}^{n+1}C_r$ for $1 \le r…Preview
+Show 8 questions8 questions
  1. Example 1Expand $(x+2)^5$ using the binomial theorem.Free
  2. Example 2Expand $(2x-3y)^4$ using the binomial theorem.Free
  3. Example 3Using the binomial theorem, prove that $9^n - 8n - 1$ is divisible by $64$ for every positive integer $n$.Free
  4. Example 4Find the coefficient of $x^5$ in the expansion of $(x+3)^8$.Preview
  5. Example 5Find the middle term in the expansion of $\left(\dfrac{x}{2}+2y\right)^8$.Preview
  6. Example 6Find the middle terms in the expansion of $\left(x-\dfrac{1}{x}\right)^7$.Preview
  7. Example 7Using the binomial theorem, find the value of $(1.02)^6$ correct to $4$ decimal places.Preview
  8. Example 8Using the binomial theorem, show that $(1.1)^{10000}$ is greater than $1000$.Preview
+Show 5 questions5 questions
  1. Q9Expand $(x+1)^6$ using the binomial theorem.Free
  2. Q10Expand $(2a-b)^5$ using the binomial theorem.Free
  3. Q11Expand $\left(\dfrac{x}{2}+\dfrac{2}{x}\right)^4$, $x \neq 0$.Preview
  4. Q12Using the binomial theorem, evaluate $(102)^4$.Preview
  5. Q13Expand $(1+x)^4 - (1-x)^4$ and simplify the result.Preview
+Show 5 questions5 questions
  1. Q14Find the coefficient of $x^4$ in the expansion of $(x+2)^9$.Free
  2. Q15Find the $7$th term in the expansion of $(3x-y)^{10}$.Free
  3. Q16Find the coefficient of $x^7$ in the expansion of $\left(x^2+\dfrac{3}{x}\right)^{11}$.Preview
  4. Q17Find the term independent of $x$ in the expansion of $\left(x-\dfrac{2}{x^2}\right)^9$.Preview
  5. Q18If the coefficients of $x^2$ and $x^3$ in the expansion of $(3+ax)^9$ are equal, find the value of $a$.Preview
+Show 4 questions4 questions
  1. Q19Find the middle term in the expansion of $\left(x+\dfrac{1}{x}\right)^{10}$.Free
  2. Q20Find the middle terms in the expansion of $(2x-3y)^7$.Free
  3. Q21Find the middle term in the expansion of $\left(3-\dfrac{x}{3}\right)^6$.Preview
  4. Q22Show that the middle term in the expansion of $(1+x)^{2n}$ is $^{2n}C_n\, x^n$.Preview
+Show 3 questions3 questions
  1. Q23Using the binomial theorem, evaluate $(1.1)^5$ correct to $3$ decimal places.Free
  2. Q24Using the binomial theorem, show that $11^n - 10n - 1$ is divisible by $100$ for every positive integer $n$.Preview
  3. Q25Using the binomial theorem, show that $(1.2)^{4000} > 800$.Preview