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I have a past exam with the solution here. But I can't understand how they combined and simplified the terms at the last line. Can anyone explain in a little more detail? Thanks.

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1 Answer 1

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What are you having trouble understanding?

  • The $\sqrt 5$ comes from dropping the square brackets and distributing the $\sqrt 5$ through.
  • The $\frac{1}{10}(x - 3)$ term becomes the value for $n =1$ in the sigma notation (note that $n$ started at $2$ before).
  • In the numerator, the $(-1)^{n+1}$ comes from two lines up. It was accidently dropped from the penultimate line, but is restored in the final expression.
  • $1\cdot2\cdot 3\cdots(2n-2)$ is the definition of $(2n-2)!$
  • In the denominator, $2\cdot4\cdot 6\cdots(2n-2) = (2\cdot1)(2\cdot2)(2\cdot3)\cdots(2(n-1)) = 2^{n-1}(n-1)!$ multiply in the other $2^n$ that is also in the denominator to get $2^{2n-1}(n-1)!$
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  • $\begingroup$ Thank you! It helped alot. $\endgroup$
    – Mike
    Nov 8, 2015 at 23:38

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