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Mathematik-Online problems:

Problem 29: Geometric Sum Formula, Induction, Binomial Coefficients

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
a)
Use induction to show the geometric sum formula:

$\displaystyle \sum_{k=0}^n q^{\mathit k} = \frac{1-q^{n+1}}{1-q}\,, \qquad \textrm{for~all~} n\in\mathbb{N},\ q\in\mathbb{R}\setminus\{1\}. $

b)
Show the following identity for the binomial coefficients $ {\displaystyle{\left({n\atop k}\right)}}$:

$\displaystyle \left({n+1\atop k}\right)=\left({n\atop k-1}\right)+\left({n\atop k}\right) $

(Authors: Kimmerle/Roggenkamp/Apprich/Höfert)

see also:

[Solutions]
  automatisch erstellt am 18. 10. 2004