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            • 1.

              设各项均为正数的数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}\),满足\(4{{S}_{n}}=a_{^{_{n+1}}}^{2}-4n-1\),且\({{a}_{1}}=1\),公比大于\(1\)的等比数列\(\left\{ {{b}_{n}} \right\}\)满足\({{b}_{2}}=3\),\({{b}_{1}}+{{b}_{3}}=10\).

              \((1)\)求证数列\(\left\{ {{a}_{n}} \right\}\)是等差数列,并求其通项公式;

              \((2)\)若\({{c}_{n}}=\dfrac{{{a}_{n}}}{3{{b}_{n}}}\),求数列\(\left\{ {{c}_{n}} \right\}\)的前\(n\)项和\({{T}_{n}}\);

              \((3)\)在\((2)\)的条件下,若\({{c}_{n}}\leqslant {{t}^{2}}+\dfrac{4}{3}t-2\)对一切正整数\(n\)恒成立,求实数\(t\)的取值范围.

            • 2.
              已知等差数列\(\{a_{n}\}\)的公差\(d\neq 0\),它的前\(n\)项和为\(S_{n}\),若\(S_{5}=70\),且\(a_{2}\),\(a_{7}\),\(a_{22}\)成等比数列.
              \((1)\)求数列\(\{a_{n}\}\)的通项公式;

              \((2)\)设数列\(\left\{ \dfrac{1}{{S}_{n}}\right\} \)的前\(n\)项和为\(T_{n}\),求证:\(T_{n} < \dfrac{3}{8} \).

            • 3.

              若各项均为正数的数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\),且\(2 \sqrt[]{S_{n}}=a_{n}+1 (n∈N*)\).

              \((1)\)求数列\(\{a_{n}\}\)的通项公式;

              \((2)\)若正项等比数列\(\{b_{n}\}\),满足\(b_{2}=2\),\(2b_{7}+b_{8}=b_{9}\),求\(T_{n}=a_{1}b_{1}+a_{2}b_{2}+…+a_{n}b_{n}\);

              \((3)\)对于\((2)\)中的\(T_{n}\),若对任意的\(n∈N^{*}\),不等式\(λ·(-1)^{n} < \dfrac{1}{2^{n+1}}(T_{n}+21)\)恒成立,求实数\(λ\)的取值范围.

            • 4.

              \((1)\int _{0}^{1}( \sqrt{1-{x}^{2}}+x+{x}^{3})dx \) ______      

              \((2)\)求值:\( \dfrac{\cos 20^{\circ}}{\cos 35^{\circ} \sqrt{1-\sin 20^{\circ}}} \) \(=\) ______         

              \((3)\)已知\(m\),\(n\),\(p\)表示不重合的三条直线,\(α\),\(β\),\(γ\)表示不重合的三个平面\(.\)下列说法正确的是 ______       \(.(\)写出所有正确命题的序号\()\).
              \(①\)若\(m⊥p\),\(m/\!/n\),则\(n⊥p\);
              \(②\)若\(m/\!/β\),\(n/\!/β\),\(m⊂α\),\(n⊂α\),则\(α/\!/β\);
              \(③\)若\(α⊥γ\),\(β⊥γ\),\(α∩β=m\),则\(m⊥γ\);
              \(④\)若\(α/\!/β\),\(m⊂α\),\(n⊂β\),则\(m/\!/n\).

              \((4)\)设函数\(y=f(x)\)的定义域为\(D\),若对于任意\(x_{1}\),\(x_{2}∈D\),当\(x_{1}+x_{2}=2a\)时,恒有\(f(x_{1})+f(x_{2})=2b\),则称点\((a,b)\)为函数\(y=f(x)\)图象的对称中心,研究函数\(f(x)=x^{3}+\sin x+2\)的图象的某一个对称点,并利用对称中心的上述定义,可得到\(f(-1)+f(- \dfrac{9}{10})+⋯+f(0)+⋯+f( \dfrac{9}{10})+f(1)= \)___           

            • 5.

              \(\Delta ABC\)的三个内角\(A\),\(B\),\(C\)成等差数列, 且\((A\vec{B}+A\vec{C})\cdot B\vec{C}=0\)则\(\Delta ABC\)的 形状为\((\)  \()\)

              A.钝角三角形
              B.等边三角形
              C.直角三角形
              D.等腰直角三角形
            • 6.

              设\(S_{n}\)为数列\(\{a_{n}\}\)前\(n\)项和,且\(S_{3}=7\),\(a_{1}+3\),\(a_{3}+4\)的等差中项为\(3a_{2}\).

              \((1)\)求\(a_{2}\);

              \((2)\)若\(\{a_{n}\}\)是等比数列,求\(a_{n}\).

            • 7.

              数列\(\{a_{n}\}\)满足\(a_{1}=1\),\({{a}_{n}}+\dfrac{{{a}_{n+1}}}{2{{a}_{n+1}}-1}=0\)

              \((\)Ⅰ\()\)求证:数列\(\left\{ \dfrac{1}{{{a}_{n}}} \right\}\) 是等差数列;

              \((\)Ⅱ\()\)若数列\(\{b_{n}\}\)满足\(b_{1}=2\),\(\dfrac{{{b}_{n+1}}}{{{b}_{n}}}=\dfrac{2{{a}_{n}}}{{{a}_{n+1}}}\) ,求\(\{b_{n}\}\)的前\(n\)项和\(S_{n}\).

            • 8.

              设数列\(\left\{ {{a}_{n}} \right\}\)满足\({{a}_{1}}+2{{a}_{2}}=2\),且对任意的\(n\in {{N}^{*}}\),点\({{P}_{n}}(n,{{a}_{n}})\)都有\(\overrightarrow{{{P}_{n}}{{P}_{n+1}}}=(1,2)\),则数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为__________.

            • 9. 若数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\),且满足\(a_{n}+2S_{n}S_{n-1}=0(n\geqslant 2)\),\(a_{1}= \dfrac{1}{2}\).
              \((1)\)求证:\(\left\{ \left. \dfrac{1}{S_{n}} \right. \right\}\)成等差数列;

              \((2)\)求数列\(\{a\)\({\,\!}_{n}\)\(\}\)的通项公式.

            • 10.

              已知数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\),且\(1\),\(a_{n}\),\(S_{n}\)是等差数列.

              \((\)Ⅰ\()\)求数列\(\{a_{n}\}\)的通项公式;

              \((\)Ⅱ\()\)若\(b_{n}=\log _{2}a_{n}\),设\(c_{n}=a_{n}⋅b_{n}\),求数列\(\{c_{n}\}\)的前\(n\)项和为\(T_{n}\).

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