Formula Sheet: Binomial Theorem
This chapter introduces the binomial theorem, which simplifies the expansion of binomials raised to a power. It is essential for efficiently calculating powers without repeated multiplication.
Binomial Theorem – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Binomial Theorem chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formula sheet
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
(a + b)^n = ∑(nCk)(a^(n-k))(b^k) for k = 0 to n
This is the Binomial Theorem, where n is a non-negative integer, a and b are terms, and nCk are binomial coefficients. It provides a quick way to expand binomials raised to any power.
nCk = n! / (k!(n-k)!)
Here, nCk represents the number of combinations of n items taken k at a time. This formula is crucial for calculating the coefficients in binomial expansions.
nC0 = 1 and nCn = 1
These are the base cases for binomial coefficients, indicating that there is one way to choose none or all items from a set of n items.
Pascal's Triangle
This triangular array of binomial coefficients provides an easy way to find coefficients for binomial expansions without calculating combinations.
(1 + x)^n = ∑(nCk)(x^k) for k = 0 to n
This special case of the Binomial Theorem is useful for simplifying expressions where a = 1. It shows how any power of a binomial can be expressed as a polynomial in x.
(x - y)^n = ∑(nCk)(x^(n-k))(-y)^k for k = 0 to n
This is the expansion for (x - y)^n, showcasing how negative terms affect binomial expansions.
∑(nCk) = 2^n
This equation states that the sum of the coefficients in the expansion of (1 + x)^n equals 2^n, useful for quick checks.
a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + ... + b^{n-1})
This represents a polynomial identity for simplifying expressions involving powers of two terms.
(a + b)^n = a^n + nC1 a^{n-1} b + nC2 a^{n-2} b^2 + ... + b^n
This detailed form shows the first few terms of the binomial expansion, useful for manual calculations for smaller n.
(1 + a)^n = 1 + na + (n(n-1)/2!)a^2 + ... + a^n
This provides the first few terms of the binomial expansion for small values of a, useful for approximations.
Equations
(x + 2)^6 = 6C0 x^6 + 6C1 x^5 (2) + 6C2 x^4 (2^2) + ... + 6C6 (2^6)
An example of applying the Binomial Theorem to expand (x + 2)^6, showing how each coefficient corresponds to a term in the expansion.
P(n) : (a + b)^n = ∑(nCk)a^(n-k)b^k
The statement P(n) represents the Binomial Theorem applied to any positive integer n, forming the basis for inductive proofs.
(2x + 3y)^5 = ∑(5Ck)(2x)^(5-k)(3y)^k
This shows how to expand (2x + 3y)^5 using the Binomial Theorem, where k varies from 0 to 5.
(1 + x)^n = nC0 + nC1 x + nC2 x^2 + ... + nCk x^k
This is the expanded form using specific coefficients for powers of x, emphasizing the structure of the binomial expansion.
nCk + nC(k-1) = n+1Ck
This recurrence relation shows how coefficients in Pascal's Triangle relate to each other, useful for deriving rows in the triangle.
(x - 3)^4 = ∑(4Ck)x^(4-k)(-3)^k
An application of the Binomial Theorem to show how to expand (x - 3)^4 while keeping in mind the negative term.
f(n) = 6n - 5n = 25k + 1
This shows how the Binomial Theorem can prove divisibility properties by manipulating polynomial forms.
nC1 = n, nC2 = n(n-1)/2
Common binomial coefficients for small values that are often encountered in expanded forms.
(x + 1)^5 = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1
The expansion showcasing the structure of coefficients for a practical application of the Binomial Theorem.
(3y + 2)^4 = ∑(4Ck)(3y)^(4-k)(2)^k
Applicable to show how to expand with non-integer coefficients, illustrating flexibility of the binomial expansion.
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