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Toplam Sembolü, Formülleri ve Özellikleri

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By on Bilgi Dünyası
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Toplam sembolü nasıldır? Toplam özellikleri, önemli formülleri, hakkında bilgi.

TOPLAM (∑)

∀k∈N+için ak∈R olmak üzere;

\displaystyle \sum\limits_{k=1}^{n}{{{a}_{k}}}={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...+{{a}_{n}}

ÖNEMLİ TOPLAM FORMÜLLERİ:

\displaystyle 1)\sum\limits_{k=1}^{n}{k}=1+2+...+n=\frac{n\left( n+1 \right)}{2}

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\displaystyle 2)\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}=1+3+5+...+\left( 2n-1 \right)={{n}^{2}}

\displaystyle 3)\sum\limits_{k=1}^{n}{2k}=2+4+6+...+2n=n.\left( n+1 \right)

\displaystyle 4)\sum\limits_{k=1}^{n}{{{k}^{2}}}={{1}^{2}}+{{2}^{2}}+{{3}^{2}}+...+{{n}^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}

\displaystyle 5)\sum\limits_{k=1}^{n}{{{k}^{3}}}={{1}^{3}}+{{2}^{3}}+{{3}^{3}}+...+{{n}^{3}}={{\left[ \frac{n\left( n+1 \right)}{2} \right]}^{2}}

\displaystyle 6)\sum\limits_{k=1}^{n}{{{r}^{k-1}}}=1+r+{{r}^{2}}+...+{{r}^{n-1}}=\frac{1-{{r}^{n}}}{1-r}

\displaystyle 7)1.2+2.3+3.4+...+n.\left( n+1 \right)=\frac{n.\left( n+1 \right)\left( n+2 \right)}{3}

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\displaystyle 8)1.2.3+2.3.4+3.4.5+...+n.\left( n+1 \right)\left( n+2 \right)=\frac{n.\left( n+1 \right)\left( n+2 \right)\left( n+3 \right)}{4}

\displaystyle 9)\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{n.\left( n+1 \right)}=\frac{n}{n+1}

\displaystyle 10){{\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}}^{2}}={{1}^{2}}+{{3}^{2}}+{{5}^{2}}+...+{{\left( 2n-1 \right)}^{2}}=\frac{n.\left( 2n-1 \right)\left( 2n+1 \right)}{3}

TOPLAM ÖZELLİKLERİ

\displaystyle *c\in R\Rightarrow \sum\limits_{k=1}^{n}{c}=n.c

\displaystyle *c\in R\Rightarrow \sum\limits_{k=0}^{n}{c}=\left( n+1 \right).c

\displaystyle *\sum c.{{a}_{k}}=c.\sum {{a}_{k}}

\displaystyle *\sum \left( {{a}_{k}}\pm {{b}_{k}} \right)=\sum {{a}_{k}}\pm \sum {{b}_{k}}

\displaystyle *\sum\limits_{k=m}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n-\left( m-1 \right)}{{{a}_{k+\left( m-1 \right)}}}

\displaystyle \sum\limits_{k=1}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{p}{{{a}_{k}}}+\sum\limits_{k=p+1}^{n}{{{a}_{k}}}p\in \left( 1,n \right)

\displaystyle \sum\limits_{k=m}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n}{{{a}_{k}}}-\sum\limits_{k=1}^{m-1}{{{a}_{k}}}

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\displaystyle \sum\limits_{k=p}^{n}{{{a}_{k}}}={{a}_{p}}+\sum\limits_{k=p+1}^{n}{{{a}_{k}}}


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