CHSE Odisha Class 11 Math Notes Chapter 9 Binomial Theorem

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Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 9 Binomial Theorem will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 9 Binomial Theorem

Binomial Theorem For Positive Integral Index:
For any a,b ∈ R, and n ∈ N
(a + b)n = nC0 an + nC1 an-1b + ….. nCn bn

Note:

(a) (a + b)n = an + nan-1 b + \(\frac{n(n-1)}{2 !}\) an-2b2 ….. + bn
(b) (1 + x)n = nC0 + nC1 x + nC2 x2 + ….. + nCn xn
(c) (a – b)n = nC0 an – nC1 an-1 b + nC2 an-2b2 ….. + (-1)n bn
(d) (1 – x)n = nC0 – nC1 x + nC2 x2 ….. + (-1)n xn

Some conclusions from the Binomial theorem:

  • There are (n + 1) terms in the expansion of (a + b)n
  • We can write (a + b)n = \(\sum_{r=0}^n{ }^n \mathrm{C}_r a^{n-r} b^r\) and (a – b)n = \(\sum_{r=0}^n(-1)^r{ }^n \mathrm{C}_r a^{n-r} b^r\)
  • The sum of powers of a and b in each term = n
  • As nCr = nCn-r (The coefficient of terms equidistant from the beginning and the end are equal).
  • (r + 1)th term (General term)
    = tr+1 = nCr an-rbr
  • (a + b)n + (a – b)n = 2[nC0an + nC2 an-2b2 + ….]
  • (a + b)n – (a – b)n = 2[nC1 an-1b + nC3 an-3b3 + ….]
  • (middle terms):
    ⇒ If n is even then the middle term = \(t_{\left(\frac{n+2}{2}\right)}=t_{\left(\frac{n}{2}+1\right)}\)
    ⇒ If n is odd there are two middle terms. They are = \(t_{\left(\frac{n+1}{2}\right)} \text { and } t_{\left(\frac{n+3}{2}\right)}\)
  • tr+1 from the end in the expansion of (a + b)n = tr+1 from the beginning in the expansion of (b + a)n.

CHSE Odisha Class 11 Math Notes Chapter 9 Binomial Theorem

Binomial Theorem For Any Rational Index:
If n ∈ Q and x ∈ R such that |x| < 1 then (1 + x)n = 1 + nx + \(\frac{n(n-1)}{2 !} x^2\) + \(\frac{n(n-1)(n-2)}{3 !} x^3+\ldots .\)

Note:

(1) (1 + x)-1 = 1 – x + x2 – x3 + …..
(2) (1 – x)-1 = 1 + x + x2 + …..
(3) (1 + x)-2 = 1 – 2x + 3x2 – 4x3 + …..
(4) (1 – x)-2 = 1 + 2x + 3x2 + 4x3 + …..

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