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#1
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S3/2 i2(1/2)i i=0 ·ÓÍÂèÒ§äè֧ä´é¤ÓµÍºÍÂèÒ§¹Õé -6[(1/2k2+k+3/2)(1/2)k-3/2] |
#2
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#3
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$$ \sum_{i=0}^{k-1}\frac{3}{2}i^2\bigg(\frac{1}{2}\bigg)^i =\frac{3}{2} \sum_{i=0}^{k-1}i^2\bigg(\frac{1}{2}\bigg)^i $$
$$ \sum_{i=0}^{k-1}\frac{3}{2}i^2\bigg(\frac{1}{2}\bigg)^i = \frac{3}{2}\Bigg(0+1(\frac{1}{2})+4(\frac{1}{2})^2+\dots+(k-1)^2(\frac{1}{2})^{k-1}\Bigg)$$ $$ãËé\quad S_k=\Bigg(1(\frac{1}{2})+4(\frac{1}{2})^2+\dots+(k-1)^2(\frac{1}{2})^{k-1}\Bigg)\quad ...(1)$$ $$\frac{1}{2}S_k = \Bigg(1(\frac{1}{2})^2+4(\frac{1}{2})^3+\dots+(k-1)^2(\frac{1}{2})^k\Bigg) \quad ...(2)$$ \[ (1)-(2);\frac{1}{2}S_k=\Bigg(\frac{1}{2}+3(\frac{1}{2})^2+5(\frac{1}{2})^3+\dots+(2k-3)(\frac{1}{2})^{k-1}-(k-1)^2(\frac{1}{2})^k\Bigg) \] \[\frac{1}{2}S_k=\Bigg(\bigg(\frac{1}{2}+3(\frac{1}{2})^2+5(\frac{1}{2})^3+\dots+(2k-3)(\frac{1}{2})^{k-1}\bigg)-(k-1)^2(\frac{1}{2})^k\Bigg) \] $$ãËé \quad S_g=\bigg(\frac{1}{2}+3(\frac{1}{2})^2+5(\frac{1}{2})^3+\dots+(2k-3)(\frac{1}{2})^{k-1}\bigg)\quad ...(3)$$ $$\frac{1}{2}S_g=(\frac{1}{2})^2+3(\frac{1}{2})^3+5(\frac{1}{2})^4+\dots+(2k-3)(\frac{1}{2})^{k} \quad ...(4)$$ $$(3)-(4);\frac{1}{2}S_g=\bigg(\frac{1}{2}+2(\frac{1}{2})^2+2(\frac{1}{2})^3+\dots+2(\frac{1}{2})^{k-1}-(2k-3)(\frac{1}{2})^k\bigg)$$ $$\frac{1}{2}S_g=\bigg(\frac{1}{2}+2\Big((\frac{1}{2})^2+(\frac{1}{2})^3+(\frac{1}{2})^4+\dots+(\frac{1}{2})^{k-1}\Big)-(2k-3)(\frac{1}{2})^k\bigg)$$ $$\frac{1}{2}S_g=\Bigg(\frac{1}{2}+2\bigg(\frac{\frac{1}{4}\big(1-(\frac{1}{2})^{k-2}\big)}{1-1/2}\bigg)-(2k-3)(\frac{1}{2})^k\Bigg)$$ $$\frac{1}{2}S_g=\bigg(\frac{1}{2}+1-4(\frac{1}{2})^k-(2k-3)(\frac{1}{2})^k\bigg)$$ $$S_g=\bigg(3-(4k+2)(\frac{1}{2})^k\bigg) $$ $$S_k=2\Bigg(\bigg(3-(4k+2)(\frac{1}{2})^k\bigg)-(k-1)^2(\frac{1}{2})^k\Bigg)$$ $$S_k=2\Big(3-(k^2+2k+3)(\frac{1}{2})^k\Big)$$ $$\sum_{i=0}^{k-1}\frac{3}{2}i^2\bigg(\frac{1}{2}\bigg)^i = \frac{3}{2}S_k$$ $$\sum_{i=0}^{k-1}\frac{3}{2}i^2\bigg(\frac{1}{2}\bigg)^i = -3\bigg((k^2+2k+3)(\frac{1}{2})^k-3\bigg)$$
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