What is the binomial expansion of ${(2 x + \frac{1}{x})}^{7}$ $(2x+1/x)^7$ ?

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What is the binomial expansion of ${(2 x + \frac{1}{x})}^{7}$ $(2x+1/x)^7$ ?

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Answer 1 · 2015-05-26T18:01:33

The 8th row of Pascal's triangle goes:

$1$ $1$ , $7$ $7$ , $21$ $21$ , $35$ $35$ , $35$ $35$ , $21$ $21$ , $7$ $7$ , $1$ $1$

Multiply this by descending powers of 2:

$128$ $128$ , $64$ $64$ , $32$ $32$ , $16$ $16$ , $8$ $8$ , $4$ $4$ , $2$ $2$ , $1$ $1$

To get the sequence:

$128$ $128$ , $448$ $448$ , $672$ $672$ , $560$ $560$ , $280$ $280$ , $84$ $84$ , $14$ $14$ , $1$ $1$

So:

${(2 x + \frac{1}{x})}^{7}$ $(2x+1/x)^7$

$= 128 x^{7} + 448 x^{5} + 672 x^{3} + 560 x + \frac{280}{x} + \frac{84}{x^{3}} + \frac{14}{x^{5}} + \frac{1}{x^{7}}$ $=128x^7 + 448x^5 + 672x^3 + 560x + 280/x + 84/x^3 + 14/x^5 + 1/x^7$

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What is the binomial expansion of ${(2 x + \frac{1}{x})}^{7}$ $(2x+1/x)^7$ ?

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