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Given that \(A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,\) find \(A_{19} + A_{20} + \cdots + A_{98}.\)

 Aug 19, 2019
 #1
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+1

∑[k*(k - 1) /2 *cos (k*(k - 1) *pi /2), k, 19, 98] = - 40
A1 - A18 = (0, -1, -3, 6, 10, -15, -21, 28, 36, -45, -55, 66, 78, -91, -105, 120, 136, -153)
A19 - A98 =(-171, 190, 210, -231, -253, 276, 300, -325, -351, 378, 406, -435, -465, 496, 528, -561, -595, 630, 666, -703, -741, 780, 820, -861, -903, 946, 990, -1035, -1081, 1128, 1176, -1225, -1275, 1326, 1378, -1431, -1485, 1540, 1596, -1653, -1711, 1770, 1830, -1891, -1953, 2016, 2080, -2145, -2211, 2278, 2346, -2415, -2485, 2556, 2628, -2701, -2775, 2850, 2926, -3003, -3081, 3160, 3240, -3321, -3403, 3486, 3570, -3655, -3741, 3828, 3916, -4005, -4095, 4186, 4278, -4371, -4465, 4560, 4656, -4753)>>Total = - 40

Note: somebody should ckeck this. Thanks.

 Aug 19, 2019
 #2
avatar+23861 
+3

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

 

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

 

laugh

 Aug 20, 2019

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