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Aug 20, 2019
 #2
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Aug 20, 2019
 #2
avatar+26364 
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Given that \(A_k = \dfrac {k(k - 1)}2\cos\left( \dfrac {k(k - 1)\pi}2 \right)\),
find \(A_{19} + A_{20} + \cdots + A_{98}\).

 

\(\begin{array}{|rcll|} \hline &&\mathbf{ A_{19} + A_{20} + \cdots + A_{98}} \quad | \quad \small{98-18 = 80 \text{ terms}} \\\\ &=& \sum \limits_{k=19}^{98} \dfrac {k(k - 1)}2\cos\left( \dfrac {k(k - 1)\pi}2 \right) \\\\ &=&\dfrac{1}{2} \sum \limits_{k=19}^{98} k(k - 1)\cos\left( \dfrac {k(k - 1)\pi}2 \right) \\\\ &=&\dfrac{1}{2} \sum \limits_{k=19}^{98} k(k - 1)\cos(n\pi) \quad | \quad n = \dfrac{k(k - 1)}2 \\ && \boxed{\cos(n\pi)=\cos^n(\pi)\quad | \quad \cos(\pi) = -1 \\\cos(n\pi) = (-1)^{n} } \\ &=&\dfrac{1}{2} \sum \limits_{k=19}^{98} k(k - 1)(-1)^{n} \\\\ &=&\mathbf{\dfrac{1}{2} \sum \limits_{k=19}^{98} k(k - 1)(-1)^{\left(\dfrac{k(k - 1)}2\right)} } \\ \hline &=& \small{\dfrac{1}{2}\Big( -18\cdot 19+19\cdot 20+20\cdot 21-21\cdot 22-22\cdot 23+23\cdot 24+24\cdot 25-25\cdot 26 +- \ldots } \\ && \small{-94\cdot 95+95\cdot 96 + 96\cdot 97-97\cdot 98 \Big)} \quad | \quad \small{80 \text{ terms}} \\ &=& \small{\dfrac{1}{2}\Big( 19(-18+20)+21(20-22)+23(-22+24)+25(24-26) + \ldots }\\ && \small{+95(-94+96) +97(96-98)\Big)} \quad | \quad \small{40 \text{ terms}} \\ &=& \small{\dfrac{1}{2}\Big( 2\cdot 19 -2\cdot 21+2\cdot 23-2\cdot 25 +- \ldots +2\cdot 95 -2\cdot 97 \Big)} \quad | \quad \small{40 \text{ terms}} \\ &=& \small{\dfrac{2}{2}( 19 - 21+ 23- 25 +- \ldots + 95 - 97 )} \quad | \quad \small{40 \text{ terms}} \\ &=& \underbrace{ \underbrace{19-21}_{=-2} + \underbrace{23-25}_{=-2} + \underbrace{27-29}_{=-2} +- \ldots + \underbrace{91-93}_{=-2} + \underbrace{95-97}_{=-2}}_{ 20 \text{ terms} } \\ &=& -2\cdot 20 \\ &=& \mathbf{-40} \\ \hline \end{array}\)

 

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Aug 20, 2019

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