Binomial Theorem Solutions

Binomial Theorem for Positive Index

(a + x)ⁿ = nC₀aⁿ + nC₁a^n-1x + nC₂a^n-2x² + ... + nC_ra^n-rx^r + ... + nC_nxⁿ

where nC₀, nC₁, nC₂, ..., nC_n are called Binomial coefficients. The value of nC_r is defined as

nC_r = n! / [r! (n - r)!] = n(n-1)(n-2)...(n-r+1) / (1 × 2 × 3 × ... × r)

Observations

1. There are (n+1) terms in the expansion of (a + x)ⁿ.
2. Sum of powers of x and a in each term in the expansion of (a + x)ⁿ is constant and equal to n.
3. The general term in the expansion of (a + x)ⁿ is the (r+1)^th term, given as t_r+1 = nC_ra^n-rx^r.
4. The binomial coefficients in the expansion of (a + x)ⁿ equidistant from the beginning and the end are equal. That means nC_r is equal to nC_n-r.

Middle Terms

(a) When n is even
Clearly in this case we have only one middle term namely t_{n/2 + 1}. Thus middle term in the expansion of (a + x)ⁿ will be ⁿC_n/2 a^n/2x^n/2 term.

(b) When n is odd
Clearly in this case we have two middle terms namely t_(n+1)/2 and t_(n+3)/2. That means the middle terms in the expansion of (a + x)ⁿ are ⁿC_(n-1)/2a^(n+1)/2x^(n-1)/2 and ⁿC_(n+1)/2a^(n-1)/2x^(n+1)/2.

Greatest Term

Greatest Term

If t_r and t_r+1 be the r^th and (r + 1)^th term in the expansion of (1 + x)ⁿ, then

t_r+1 / t_r = ⁿC_r X^r / ⁿC_r-1 X^n-r= n - r + 1 /r = x

Let numerically, t_r+1 be the greatest term in the above expansion. Then t_r+1 ≥ t_r or

tr+1tr ≥ 1 ⇒n - r + 1 r|x| ≥ 1⇒r ≤ (n+1)|x| 1+|x|

Now shifting values of n and x in (2), we get r ≤ m + f or r ≤ m

Where m is a positive integer, f is a fraction such that 0 ≤ f < 1.
Now if f = 0 then t_m+1 and t_m both the terms will be numerically equal and greatest while if f ≠ 0, then t_m+1 is the greatest term of the binomial expansion.
i.e. to find the greatest term (numerically) in the expansion of (1 + x)ⁿ.

1. Calculate m = (n + 1) |x|
   —————————
   (1 + |x|)
2. If m is integer, then t_m and t_m+1 are equal and are greatest term.
3. If m is not integer, then t_[m]+1 is the greatest term (where [.] denotes the greatest integer function).

PROPERTIES OF BINOMIAL COEFFICIENTS

For the sake of convenience, the coefficients nC₀ , nC₁ , ..., nC_r ,..., nC_n are usually denoted by C₀ , C₁ , ..., C_r , ..., C_n respectively

1. C₀ + C₁ + C₂ + ... + C_n = 2ⁿ
2. C₀ − C₁ + C₂ − ... + (−1)ⁿC_n = 0
3. C₀ + C₂ + C₄ + ... = C₁ + C₃ + C₅ + ... = 2ⁿ⁻¹
4. ⁿC_r₁ = ⁿC_r₂ ⇒ r₁ = r₂ or r₁ + r₂ = n
5. nC_r + nC_r-1 = n+1C_r
6. r·nC_r = n·n−1C_r-1
7. ⁿC_r/r+1 = ⁿ⁺¹C_r+1/n+1

(1 + x)ⁿ = C₀ + C₁x + C₂x² + ... + Cₙxⁿ ...(1)
and ( x + 1 )ⁿ = C₀xⁿ + C₁x^n-1 + ... + Cₙ ...(2)
multiplying (1) and (2), we get
(1 + x )²ⁿ = (C₀ + C₁x + C₂x² + ... + Cₙxⁿ)(C₀xⁿ + C₁x^n-1 + ... + Cₙ)
Equating the coefficient of xⁿ⁺¹, we get

C₀C₁ + C₁C₂ + ... + C_n-1Cₙ = 2nC_n+1 = (2n)!
(n + 1)! (n - 1)!

BINOMIAL THEOREM FOR ANY INDEX

(1 + x)ⁿ = 1 + nx + ^{n(n - 1)}⁄_2! x² + … + ^{n(n - 1)…(n - r + 1)}⁄_r! x^r + … terms up to ∞

Observations

1. Expansion is valid only when −1 < x < 1
2. nC_r can not be used because it is defined only for natural numbers, so nC_r will be written as ^{n(n - 1)…(n - r + 1)}⁄_r!.
3. As the series never terminates, the number of terms in the series is infinite.
4. General term of the series (1 + x)⁻ⁿ = T_r+1 = (−1)^{rn(n+1)(n+2)…(n+r-1)}⁄_r! x^r

Important Expansions

1. (1 + x)^-1 = 1 − x + x² − x³ + ... + (−1)^rx^r + ...
2. (1 − x)^-1 = 1 + x + x² + x³ + ... + x^r + ...
3. (1 + x)^-2 = 1 − 2x + 3x² − 4x³ + ... + (−1)^rr(r + 1)x^r + ...
4. (1 − x)^-2 = 1 + 2x + 3x² + 4x³ + ... + (r + 1)x^r + ...
5. (1 + x)^-3 = 1 − 3x + 6x² − 10x³ + ... + (−1)^{r(r + 1)(r + 2)}/_r! x^r + ...
6. (1 − x)^-3 = 1 + 3x + 6x² + 10x³ + ... + ^{(r + 1)(r + 2)}/_r! x^r + ...
7. (1 − x)^-p/q = 1 + ^p/_1! (x/q) + ^{p(p + q)}/_2! (x/q)² + ...

Multinomial Expansion

In the expansion of (x₁ + x₂ + ... + x_n)^m where m, n ∈ ℕ and x₁, x₂, ..., x_n are independent variables, we have

1. Total number of terms in the expansion = C(m + n - 1, n - 1)
2. Coefficient of x₁^r₁ x₂^r₂ x₃^r₃ ... x_n^r_n (where r₁ + r₂ + ... + r_n = m, r_i ∈ ℕ ∪ {0}) is
   ^m!⁄_{r1! r2! ... rn!}

Sum of all the coefficients is obtained by putting all the variables x_i equal to 1 and it is equal to n^m.

Formulas and Concepts at a Glance

1. (a + x)ⁿ = ⁿC₀aⁿ + ⁿC₁a^n-1x + ⁿC₂a^n-2x² + ... + ⁿC_ra^n-rx^r + ... + ⁿC_nxⁿ = ∑_r=0ⁿⁿC_rx^r
2. (a + x)ⁿ + (a - x)ⁿ = ∑_r=0ⁿⁿC_rx^r[1 + (-1)^r] = 2(ⁿC₀aⁿ + ⁿC₂a^n-2x² + ...)
3. (a + x)ⁿ - (a - x)ⁿ = ∑_r=0ⁿⁿC_rx^r[1 - (-1)^r] = 2(ⁿC₁a^n-1x + ⁿC₃a^n-3x³ + ...)
4. C₀ + C₁ + C₂ + ... + C_n = 2ⁿ
5. C₀ - C₁ + C₂ - ... = 0
6. C₀ + C₂ + C₄ + ... = C₁ + C₃ + C₅ + ... = 2^n-1
7. ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r
8. ⁿC_r = [(n - r + 1) / r] ⁿC_r-1
9. rⁿC_r = n^n-1C_r-1
10. ⁿC_r / (r + 1) = ⁿ⁺¹C_r+1 / (n + 1)
11. There are (n + 1) terms in expansion.
12. Binomial coefficients of terms equidistant from beginning and end are equal.
13. The general term of the expansion is ⁿC_ra^n-rx^r, this is in fact the (r+1)th term from the beginning.
14. If n is even, there is only one middle term namely (n/2 + 1)th and is equal to ⁿC_n/2a^n/2x^n/2.
15. If n is odd, there are two middle terms namely (n+1)/2 th and (n+3)/2 th and are equal to ⁿC_(n-1)/2a^(n+1)/2x^(n-1)/2 and ⁿC_(n+1)/2a^(n-1)/2x^(n+1)/2 respectively.
16. Binomial coefficient of middle term is the greatest binomial coefficient occurring in the expansion.