Binomial Theorem Formulas

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Binomial Theorem Formulae List

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1. Binomial Theorem for positive Integral Index

(x + a)n = nC0xna0 + nC1 xn-1 a + nC2xn-2 a2 + …… + nCrxn-r ar+ …… + nCnxan

2. General term

(r + 1)th terms is called general term
Tr+1 = nCrxn-r ar

3. Deductions of Binomial Theorem

(i) (1 + x)n = nC0 + nC1x + nC2x2 + nC3x3 + …….. + nCrxr + ……… + nCnxn
which is the standard form of binomial expansion.
General term = (r + 1)th term:
Tr+1 = nCrxr = $$\frac{n(n-1)(n-2) \ldots .(n-r+1)}{r !}$$.xr

(ii) (1 – x)n = nC0nC1x + nC2x2nC3x3 + …….. + (-1)r nCrxr + ……… + (-1)n nCnxn General term = (r + 1)th term:
Tr+1 = (-1)r. nCrxr = (-1)r. $$\frac{n(n-1)(n-2) \ldots .(n-r+1)}{r !}$$.xr

4. Number of terms in the expansion of (x + y + z)n

n+2C2 = $$\frac{(n+1)(n+2)}{2}$$
Number of terms in the expansion of (x1 + x2 + x3 + …. + xk)n are n+k-1Ck-1 when x1, x2, x3 ………. xk all are different and can not be solved.

5. Middle term in the expansion of (x + a)n

• If n is even then middle term = $$\left(\frac{\mathrm{n}}{2}+1\right)^{\mathrm{th}}$$ term.
• If n is odd then middle terms are = $$\left(\frac{n+1}{2}\right)^{t h}$$ and $$\left(\frac{n+3}{2}\right)^{t^{\prime \prime}}$$ term. Binomial coefficient of middle term is the greatest Binomial coefficient.

6. To determine a particular term in the expansion

In the expansion of $$\left(x^{\alpha} \pm \frac{1}{x^{\beta}}\right)^{n}$$, if xm occurs in Tr+1, then r is given by
nα – r (α + β) = m ⇒ r = $$\frac{n \alpha-m}{\alpha+\beta}$$ and the term which is independent of x then nα – r (α + β) = 0 ⇒ r = $$\frac{n \alpha}{\alpha+\beta}$$

7. To find a term from the end in the expansion of (x + a)n

Tr(E) = Tn-r+2(B)

8. Binomial coefficients & their properties

In the expansion of (1 + x)n = C0 + C1x + C2x2 + …… + Crxr + ….+ Cnxn where C0 = 1, C1 = n, C2 = $$\frac{n(n-1)}{2 !}$$

• C0 + C1 + C2 + ……. + Cn = 2n
• C0 – C1 + C2 – C3 + …….. = 0
• C0 + C2 + ……… = C1 + C3 + …….. = 2n-1
• C02 + C12 + C22 + ……… + Cn2 = $$\frac{2 n !}{n ! n !}$$
• $$C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots .+\frac{C_{n}}{n+1}=\frac{2^{n+1}-1}{n+1}$$
• $$C_{0}-\frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4}+\ldots+\frac{(-1)^{n} \cdot C_{n}}{n+1}=\frac{1}{(n+1)}$$

9. Greatest term in the expansion of (x + a)n

(i) The term in the expansion of (x + a)n of greatest coefficient

(ii) The greatest term
= $$\left\{\begin{array}{l}T_{p} \& T_{p+1}, \text { when } \frac{(n+1) a}{x+a}=p \in Z \\ T_{q+1}, \text { when } \frac{(n+1) a}{x+a} \notin Z \text { and } q<\frac{(n+1) a}{x+a}<q+1 \end{array}\right.$$
Note: Here take only positive values |x| and |a|

10. Binomial Theorem when “n” is any index

(1 + x)n = 1 + nx + $$\begin{array}{c} \frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{3 !} \mathrm{x}^{3}+\ldots \ldots . \\\ldots .+\frac{\mathrm{n}(\mathrm{n}-1) \ldots \ldots .(\mathrm{n}-\mathrm{r}+1)}{\mathrm{r} !} \mathrm{x}^{\mathrm{r}}+\ldots \ldots \infty \end{array}$$
General term: Tr+1 = $$\frac{n(n-1)(n-2) \ldots(n-r+1)}{r !} \cdot x^{r}$$

11. Some important expansions

• (1 -x)-1 = 1 + x + x2 + x3 + …….. + xr + …….. General term Tr+1 = xr
• (1 + x)-1 = 1 – x + x2 – x3 + …. (-x)r + …….. General term Tr+1 = (-x)r
• (1 – x)-2 = 1 + 2x + 3x2 + 4x3 + …….. + (r + 1) xr + ………. General term Tr+1 = (r + 1) xr
• (1 + x)-2 = 1 – 2x + 3x2 – 4x3 + …….. + (r + 1) (-x)r + ………. General term Tr+1 = (r + 1) (-x)r.

12. If ($$\sqrt{\mathrm{P}}$$ + Q)n = I + f where I and n are the integers, n being odd, and 0 ≤ f ≤ 1 then (I + f)f = kn, where P – Q2 = k > 0 and $$\sqrt{\mathrm{P}}$$ – Q < 1

13. Multinomial Expansion:

If n ∈ N then the general terms of the multinomial expansion (x1 + x2 + …….. + xk)n is