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等差数列求和$$ S_n= \dfrac{n(a_1+a_n)}{2}=na_1+\dfrac{n(n-1)}{2}d $$
等比数列求和$$ S_n=\left\{\begin{matrix} na_1 & (q=1)\\ & \\ \dfrac{a_1(1-q^n)}{1-q}=\dfrac{a_1-a_nq}{1-q} & (q \ne 1)\end{matrix}\right. $$
$$ \sum_{k=1}^{n}k = \dfrac{1}{2}n(n+1) $$
$$ \sum_{k=1}^{n}k^2 = \dfrac{1}{6}n(n+1)(2n+1) $$
$$ \sum_{k=1}^{n}k^3 = [\dfrac{1}{2}n(n+1)]^2 $$
1、$$ if\ geometric\ sequence\{a_n\},S_n=2^n-1,find\ a_1^2+a_2^2+\cdots + a_n^2=? $$
1、$$ \sum_{k=1}^{n} (2k-1)x^{k-1} $$
2,$$a_n=2^n,find\ S_n=\sum_{k=1}^{n}(n+1-k)a_k $$
1、$$ \sum_{k=1}^{89} sin ^2 k^{\circ} $$
2、$$C_n^0\cdot 0+C_n^1\cdot 1+\cdots+C_n^n\cdot n$$
3、$$ \sum_{k=1}^{10} \dfrac{k^2}{k^2+(11-k)^2} $$
4、$$ \sum_{k=1}^{179} cosk^\circ $$
1、$$ \sum_{k=1}^{n}(\dfrac{1}{a^{k-1}} +3k-2 ) $$
2、$$ \sum_{k=1}^{n}[k(k+1)(2k+1)] $$
3、$$ 1\dfrac{1}{2}+3\dfrac{1}{4}+5\dfrac{1}{8}+ 7\dfrac{1}{16}+ \cdots ,find\ S_n= $$
$$a_n=f(n+1)-f(n)$$
$$\dfrac{sin 1^\circ }{cosn^\circ cos(n+1)^\circ }=tan(n+1)^\circ -tan n^\circ $$
$$\dfrac{1}{n(n+1)}=\dfrac{1}{n}-\dfrac{1}{n+1}$$
$$\dfrac{(2n)^2}{(2n-1)(2n+1)}=1+\dfrac{1}{2}(\dfrac{1}{2n-1}-\dfrac{1}{2n+1})$$
$$\dfrac{1}{n(n+1)(n+2)}=\dfrac{1}{2}[\dfrac{1}{n(n+1)}-\dfrac{1}{(n+1)(n+2)}]$$
$$\dfrac{n+2}{n(n+1)}\cdot \dfrac{1}{2^n}=\dfrac{2(n+1)-n}{n(n+1)}\cdot\dfrac{1}{2^n}=\dfrac{1}{n\cdot 2^{n-1}}-\dfrac{1}{(n+1)2^n}$$
$$\dfrac{1}{(An+B)(An+C)}=\dfrac{1}{C-B}(\dfrac{1}{An+B}-\dfrac{1}{An+C})$$
$$\dfrac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}$$
1, $$\sum_{k=1}^{n}\dfrac{1}{k(k+1)} $$
2、$$ \sum_{k=1}^{n}\dfrac{1}{(k+1)(k+3)} $$
3、$$ \dfrac{1}{1}+\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\cdots+\dfrac{1}{1+2+3+\cdots +n} $$
4,$$ \sum_{k=1}^{n}\dfrac{sin1^\circ }{cosk^\circ cos(k+1)^\circ} $$
5,$$ \sum_{k=1}^{n}\dfrac{4k^2}{4k^2-1} $$
6,$$ \sum_{k=1}^{n}\dfrac{1}{k(k+1)(k+2)} $$
7,$$ \sum_{k=1}^{n}\dfrac{k+2}{k(k+1)2^k} $$
8,$$ \sum_{k=1}^{n}\dfrac{1}{(3k+1)(3k+5)} $$
9,$$ \sum_{k=1}^{n}\dfrac{1}{\sqrt{k}+\sqrt{k+1}} $$
1、$$ {a_n}\ is \ geometric\ sequence,a_5a_6=9,find\ \sum_{k=1}^{10} log_3{a_k} $$
2,$$\sum_{k=1}^{n}(-1)^{k-1}k $$
3,$$ \sum_{k=1}^{n}(-1)^{k}k^2 $$
1、$$ 1+11+111+1111+\cdots (tips:\ 999 = 10^k-1)$$
2、$$ a_n=\dfrac{8}{(n+1)(n+3)},find\ \sum_{n=1}^{\infty } (n+1)(a_n-a_{n+1}) $$
提示 $$(n+1)(a_n-a_{n+1})=\cdots=8[\dfrac{1}{(n+2)(n+4)}+\dfrac{1}{(n+3)(n+4)}]=4(1/(n+2-1/(n+4)))+8(1/(n+3)-1/(n+4))$$