Lesson 10 of 10 · 15 min
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Expansion, general term and middle terms in English
NCERT binomial theorem with exercises
Must-know facts
18 facts
- 01(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, r from 0 to n, for a positive integer n.
- 02The expansion has n + 1 terms.
- 03Power of a falls from n to 0, power of b rises from 0 to n; they always add to n.
- 04Binomial coefficients nCr form Pascal's triangle; row n is nC0 … nCn.
- 05Pascal's rule nCr + nC(r − 1) = (n + 1)Cr builds each row from the one above.
- 06nCr = nC(n − r), so every row is symmetric.
- 07General term Tᵣ₊₁ = nCr aⁿ⁻ʳ bʳ; the (r + 1)th term has b to the power r.
- 08(x − y)ⁿ has alternating signs; the term with yʳ carries (−1)ʳ.
- 09(1 + x)ⁿ = nC0 + nC1 x + … + nCn xⁿ.
- 10nC0 + nC1 + … + nCn = 2ⁿ.
- 11nC0 − nC1 + nC2 − … + (−1)ⁿ nCn = 0.
- 12Sum of the coefficients of any expansion: put the variable equal to 1.
- 13Largest binomial coefficient of row n: the middle one, nC(n/2) for even n.
- 14Row 6: 1, 6, 15, 20, 15, 6, 1, total 64.
- 15(x + 2)⁶ = x⁶ + 12x⁵ + 60x⁴ + 160x³ + 240x² + 192x + 64.
- 1698⁵ = (100 − 2)⁵ = 9039207968.
- 176ⁿ − 5n leaves remainder 1 on division by 25.
- 18Pascal's triangle was known in India as Meru Prastara, given by Pingla.
Common traps
Where marks are lost
Formulas
10 to know
Binomial theorem
(a + b)ⁿ = nC0 aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + … + nCn bⁿ
n a positive integer; n + 1 terms.
Sigma form
(a + b)ⁿ = Σ nCk aⁿ⁻ᵏ bᵏ, k = 0 to n
a⁰ = b⁰ = 1.
Binomial coefficient
nCr = n!/(r!(n − r)!)
nC0 = nCn = 1.
General term
Tᵣ₊₁ = nCr aⁿ⁻ʳ bʳ
The (r + 1)th term, 0 ≤ r ≤ n.
Pascal's rule
nCr + nC(r − 1) = (n + 1)Cr
How each row of the triangle is built.
Symmetry
nCr = nC(n − r)
Terms equally far from the two ends have equal binomial coefficients.
Difference
(x − y)ⁿ = Σ (−1)ᵏ nCk xⁿ⁻ᵏ yᵏ
Signs alternate.
One plus x
(1 + x)ⁿ = nC0 + nC1 x + nC2 x² + … + nCn xⁿ
Coefficient of xʳ is nCr.
Row sum
nC0 + nC1 + … + nCn = 2ⁿ
Put x = 1 in (1 + x)ⁿ.
Alternating sum
nC0 − nC1 + nC2 − … + (−1)ⁿ nCn = 0
Put x = 1 in (1 − x)ⁿ.
Key terms
12 terms
- Binomial
- An expression with exactly two terms, such as a + b or 2x − 3.
- Index
- The power n to which the binomial is raised.
- Expansion
- The power written out as a sum of separate terms.
- Binomial coefficient
- The number nCr in front of aⁿ⁻ʳbʳ in the expansion of (a + b)ⁿ.
- Pascal's triangle
- The triangle of binomial coefficients in which each inside entry is the sum of the two above it.
- Meru Prastara
- The name of the same triangular arrangement in Pingla's work.
- General term
- Tᵣ₊₁ = nCr aⁿ⁻ʳbʳ, a formula for any single term of the expansion.
- Coefficient
- The whole number (with sign) multiplying a power of the variable, which can include powers of numbers inside the binomial.
- Term independent of x
- The term in which the power of x is zero.
- Middle term
- The term halfway along the expansion; one for even n, two for odd n.
- Mathematical induction
- Proof by showing a statement true for n = 1 and showing that truth for k forces truth for k + 1.
- Sigma notation
- Σ written with limits, meaning the sum of the terms for each value of the counter.