Binomial Theorem – HSC Math

Posted In: Mathematics

In elementary algebra, the binomial theorem or binomial expansion describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial $(x+y)^n$ into a sum involving terms of the form a $ax^by^c$, where the exponents $b$ and $c$ are nonnegative integers with $b+c=n$, and the coefficient $a$ of each term is a specific positive integer depending on $n$ and $b$.

  1. Finite Binomial Series,
    i. $(a+x)^n=a^n+^nC_1a^{n-1}x+^nC_2a^{n-2}x^2+…+^nC_ra^{n-r}x^r+…+x^n$
  2. ii. $(a-x)^n=$
    $a^n-^nC_1a^{n-1}x+^nC_2a^{n-2}x^2-…+(-1)^r\ ^nC_ra^{n-r}x^r+…+(-1)^nx^n$

    iii. $(1+x)^n=1+^nC_1x+^nC_2x^2+…+^nC_rx^r+…+x^n$

    iv. $(1-x)^n=1-^nC_1x+^nC_2x^2-…+(-1)^r\ ^nC_rx^r+…+x^n$

  3. In expansion of $(a+x)^n,$ the $(r+1)^{th}$ term,
    $T_{r+1}=^nC_ra^{n-r}x^r$
  4. In case of $(a+x)^n$,
    $i.$if $n$ is even, middle term is $(n/2+1)^{th}$ term.
    $ii.$if $n$ is odd, middle terms are $({n+1}/2)^{th}$ and $({n+1}/2+1)^{th}$ term.
  5. If $^nC_x=^nC_y$ and $x≠y$ then $n=x+y$,
  6. For the expansion of $(a+x)^n$
    $i.{T_{r+1}}/{T_r}={n-r+1}/r.x/a$
    $ii.{^nC_1}/{^nC_{r-1}}={n-r+1}/r$
  7. Infinite Binomial Series,
    $i.(1+x)^{-n}=1-nx+{n(n+1)}/{2!}x^2+…+(-1)^r{n(n+1)…(n+r-1)}/{r!}x^r+……$
    $ii.(1-x)^{-n}=1+nx+{n(n+1)}/{2!}x^2+…+{n(n+1)…(n+r-1)}/{r!}x^r+……$
    $iii.(1+x)^{-1}=1-x+x^2+…+(-1)^rx^r+……$
    $iv.(1-x)^{-1}=1+x+x^2+…+x^r+……$
    $v.(1+x)^{-2}=1-2x+3x^2+…+(-1)^r(r+1)x^r+……$
    $vi.(1-x)^{-2}=1+2x+3x^2+…+(r+1)x^r+……$
    $vii.(1+x)^{-3}=1-3x+6x^2-10x^3…+(-1)^r{(r+1)(r+2)}/2x^r+……$
    $viii.(1-x)^{-3}=1+3x+6x^2+10x^3…+{(r+1)(r+2)}/2x^r+……$
  8. In expansion of $(1+x)^{-n}$, general term,
    $T_{r+1}=(-1)^r{n(n+1)…(n+r-1)}/{r!}x^r$
  9. In expansion of $(1-x)^{-n}$, general term,
    $T_{r+1}={n(n+1)…(n+r-1)}/{r!}x^r$
  10. For the expansion of $(1+x)^n=c_0+c_1x+c_2x^2+……+c_nx^n,$
    $i.c_0+c_1+c_2+…+c_n=2^n$
    $ii.c_0+c_2+c_4+…=c_1+c_3+c_5+… = 2^{n-1}$
    $iii.c_0^2+c_1^2+c_2^2+…+c_n^2={(2n)!}/{n!}$
    $iv.c_0c_n+c_1c_{n-1}+…+c_nc_0={(2n)!}/{n!n!}$
  11. $ln(1+x)=x-x^2/2+x^3/3-…….$
  12. $ln(1-x)=x+x^2/2+x^3/3+…….$

Post Tags: Algebra | HSC


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