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Prove that the coefficients of x^n in (1...

Prove that the coefficients of `x^n` in `(1+x)^(2n)` is twice the coefficient of `x^n` in `(1+x)^(2n-1)dot`

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To prove that the coefficient of \(x^n\) in \((1+x)^{2n}\) is twice the coefficient of \(x^n\) in \((1+x)^{2n-1}\), we will follow these steps: ### Step 1: Identify the Coefficient in \((1+x)^{2n}\) The coefficient of \(x^n\) in the expansion of \((1+x)^{2n}\) can be found using the binomial theorem. According to the binomial theorem, the general term \(T_{r+1}\) in the expansion of \((1+x)^m\) is given by: \[ T_{r+1} = \binom{m}{r} x^r \] ...
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Prove that he coefficient of x^n in the expansion of (1+x)^(2n) is twice the coefficient of x^(n) in the expansion of (1+x)^(2n-1)

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Knowledge Check

  • What is the coefficient of x^(n) in (x^(2) + 2x)^(n-1) ?

    A
    `(n-1)2^(n-1)`
    B
    `(n-1)xx2^(n-1)`
    C
    `(n-1)2^(n)`
    D
    `n.2(n-1)`

  • Let the coefficient of x^(n) in the expansion of (1+x)^(2n) be P and the coefficient of x^(n) in the expansion of (1+x)^(2n-1) be q , then

    A
    `P = 2q`
    B
    `2P = q`
    C
    `2P = 3q`
    D
    `3P = 2q`

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