2 ( S n − S n − 1 ) = 3 n + 3 − ( 3 n − 1 + 3 ) = 2 × 3 n − 1 = 2 a n . {\displaystyle 2(S_{n}-S_{n-1})=3^{n}+3-(3^{n-1}+3)=2\times 3^{n-1}=2a_{n}.} a n = 3 n − 1 . {\displaystyle a_{n}=3^{n-1}.} 1. {\displaystyle 1.} ( 1 ) {\displaystyle (1)} 2013 {\displaystyle 2013} ( 2 ) {\displaystyle (2)} 2017 {\displaystyle 2017} 2. {\displaystyle 2.} 2015 {\displaystyle 2015} ( I ) {\displaystyle (I)} n = 1 {\displaystyle n=1} 2 S 1 = 2 a 1 = 6 ⇒ a 1 = 3. {\displaystyle 2S_{1}=2a_{1}=6\Rightarrow a_{1}=3.} { a n } {\displaystyle \left\{a_{n}\right\}} a n = { 3 , n = 1 , 3 n − 1 , n ⩾ 2. {\displaystyle a_{n}={\begin{cases}3,&n=1,\\3^{n-1},&n\geqslant 2.\end{cases}}} ( I I ) {\displaystyle (II)} 3 b 1 = log 3 3 = 1 ⇒ b 1 = 1 3 . {\displaystyle 3b_{1}=\log _{3}3=1\Rightarrow b_{1}={\frac {1}{3}}.} 3 n − 1 ⋅ b n = n − 1 ⇒ b n = ( n − 1 ) ⋅ 3 1 − n . {\displaystyle 3^{n-1}\cdot b_{n}=n-1\Rightarrow b_{n}=(n-1)\cdot 3^{1-n}.} b n = { 1 3 , n = 1 , ( n − 1 ) ⋅ 3 1 − n , n ⩾ 2. {\displaystyle b_{n}={\begin{cases}{\frac {1}{3}},&n=1,\\(n-1)\cdot 3^{1-n},&n\geqslant 2.\end{cases}}} n − 1 {\displaystyle \color {BrickRed}n-1} T 2 {\displaystyle \color {BrickRed}T_{2}} 1 {\displaystyle \color {BrickRed}1} n {\displaystyle \color {BrickRed}n} ( A n + B ) q n − B {\displaystyle \color {BrickRed}(An+B)q^{n}-B} 1 3 {\displaystyle \color {BrickRed}{\frac {1}{3}}} T n . {\displaystyle \color {BrickRed}T_{n}.} { ( 1 A + B ) ( 1 3 ) 1 − B = 1 3 ( 2 A + B ) ( 1 3 ) 2 − B = 5 9 {\displaystyle \color {BrickRed}{\begin{cases}(1A+B)\left({\frac {1}{3}}\right)^{1}-B={\frac {1}{3}}\\(2A+B)\left({\frac {1}{3}}\right)^{2}-B={\frac {5}{9}}\end{cases}}} { A = − 1 2 , B = − 3 4 . {\displaystyle \color {BrickRed}{\begin{cases}A={\begin{smallmatrix}-\end{smallmatrix}}{\frac {1}{2}},\\B={\begin{smallmatrix}-\end{smallmatrix}}{\frac {3}{4}}.\end{cases}}} R n = ( − n − 1 2 − 3 4 ) × 3 − ( n − 1 ) + 3 4 = ( − n 2 − 1 4 ) × 3 1 − n + 3 4 . {\displaystyle \color {BrickRed}R_{n}=(-{\frac {n-1}{2}}-{\frac {3}{4}})\times 3^{-(n-1)}+{\frac {3}{4}}=(-{\frac {n}{2}}-{\frac {1}{4}})\times 3^{1-n}+{\frac {3}{4}}.} T n = R n + 1 3 = ( − n 2 − 1 4 ) × 3 1 − n + 13 12 . {\displaystyle \color {BrickRed}T_{n}=R_{n}+{\frac {1}{3}}=(-{\frac {n}{2}}-{\frac {1}{4}})\times 3^{1-n}+{\frac {13}{12}}.} T n = 1 3 + ( 2 − 1 ) × 3 1 − 2 + ( 3 − 1 ) × 3 1 − 3 + ⋅ ⋅ ⋅ + ( n − 1 ) × 3 1 − n {\displaystyle T_{n}={\frac {1}{3}}+(2-1)\times 3^{1-2}+(3-1)\times 3^{1-3}+\cdot \cdot \cdot +(n-1)\times 3^{1-n}} 1 3 T n = 1 9 + ( 2 − 1 ) × 3 1 − 3 + ( 3 − 1 ) × 3 1 − 4 + ⋅ ⋅ ⋅ + ( n − 2 ) × 3 1 − n + ( n − 1 ) × 3 − n {\displaystyle {\frac {1}{3}}T_{n}={\frac {1}{9}}+(2-1)\times 3^{1-3}+(3-1)\times 3^{1-4}+\cdot \cdot \cdot +(n-2)\times 3^{1-n}+(n-1)\times 3^{-n}} − {\displaystyle -} 2 3 T n = 2 9 + ( 2 − 1 ) × 3 1 − 2 + 3 1 − 3 + 3 1 − 4 + ⋅ ⋅ ⋅ + 3 1 − n − ( n − 1 ) × 3 − n = ( − n 3 − 1 6 ) × 3 1 − n + 13 18 . {\displaystyle {\frac {2}{3}}T_{n}={\frac {2}{9}}+(2-1)\times 3^{1-2}+3^{1-3}+3^{1-4}+\cdot \cdot \cdot +3^{1-n}-(n-1)\times 3^{-n}=(-{\frac {n}{3}}-{\frac {1}{6}})\times 3^{1-n}+{\frac {13}{18}}.} T n {\displaystyle \color {BrickRed}T_{n}} T n = ( − n 2 − 1 4 ) × 3 1 − n + 13 12 . {\displaystyle T_{n}=(-{\frac {n}{2}}-{\frac {1}{4}})\times 3^{1-n}+{\frac {13}{12}}.} ( 2 ) {\displaystyle (2)} 2016 {\displaystyle 2016} a n = S n − S n − 1 = 3 n 2 + 8 n − 3 ( n − 1 ) 2 − 8 ( n − 1 ) = 6 n + 5. {\displaystyle a_{n}=S_{n}-S_{n-1}=3n^{2}+8n-3(n-1)^{2}-8(n-1)=6n+5.} { b n } {\displaystyle \left\{b_{n}\right\}} d . {\displaystyle d.} a n = b n + b n + 1 = 2 b n + d . {\displaystyle a_{n}=b_{n}+b_{n+1}=2b_{n}+d.} a 1 = 2 b 1 + d = 11 , a 2 = 2 b 2 + d = 2 b 1 + 3 d = 17. {\displaystyle a_{1}=2b_{1}+d=11,a_{2}=2b_{2}+d=2b_{1}+3d=17.} { b 1 = 4 , d = 3. {\displaystyle {\begin{cases}b_{1}=4,\\d=3.\end{cases}}} b n = 4 + ( n − 1 ) × 3 = 3 n + 1. {\displaystyle b_{n}=4+(n-1)\times 3=3n+1.} c n = ( a n + 1 ) n + 1 ( b n + 2 ) n = 2 n + 1 × ( 3 n + 3 ) n + 1 ( 3 n + 3 ) n = ( 3 n + 3 ) × 2 n + 1 . {\displaystyle c_{n}={\frac {(a_{n}+1)^{n+1}}{(b_{n}+2)^{n}}}={\frac {2^{n+1}\times (3n+3)^{n+1}}{(3n+3)^{n}}}=(3n+3)\times 2^{n+1}.} T n = ( 3 + 3 ) × 2 2 + ( 6 + 3 ) × 2 3 + ( 9 + 3 ) × 2 4 + ⋅ ⋅ ⋅ + ( 3 n + 3 ) × 2 n + 1 . {\displaystyle T_{n}=(3+3)\times 2^{2}+(6+3)\times 2^{3}+(9+3)\times 2^{4}+\cdot \cdot \cdot +(3n+3)\times 2^{n+1}.} 2 T n = ( 3 + 3 ) × 2 3 + ( 6 + 3 ) × 2 4 + ⋅ ⋅ ⋅ + 3 n ⋅ 2 n + 1 + ( 3 n + 3 ) × 2 n + 2 . {\displaystyle 2T_{n}=(3+3)\times 2^{3}+(6+3)\times 2^{4}+\cdot \cdot \cdot +3n\cdot 2^{n+1}+(3n+3)\times 2^{n+2}.} { ( 1 A + B ) 2 1 − B = 24 , ( 2 A + B ) 2 2 − B = 96. {\displaystyle \color {BrickRed}{\begin{cases}(1A+B)2^{1}-B=24,\\(2A+B)2^{2}-B=96.\end{cases}}} { A = 12 , B = 0. {\displaystyle \color {BrickRed}{\begin{cases}A=12,\\B=0.\end{cases}}} T n = 12 n ⋅ 2 n . {\displaystyle T_{n}=12n\cdot 2^{n}.} − T n = 6 × 2 2 + 3 × ( 2 3 + 2 4 + ⋅ ⋅ ⋅ + 2 n + 1 ) − ( 3 n + 3 ) × 2 n + 2 = − 12 n ⋅ 2 n . {\displaystyle -T_{n}=6\times 2^{2}+3\times (2^{3}+2^{4}+\cdot \cdot \cdot +2^{n+1})-(3n+3)\times 2^{n+2}=-12n\cdot 2^{n}.} ( 3 ) {\displaystyle (3)} 2017 {\displaystyle 2017} d . {\displaystyle d.} S 4 = ( a 6 − 5 d ) + ( a 6 − 4 d ) + ( a 6 − 3 d ) + ( a 6 − 2 d ) = − 14 d = 14. {\displaystyle S_{4}=(a_{6}-5d)+(a_{6}-4d)+(a_{6}-3d)+(a_{6}-2d)=-14d=14.} d = − 1. {\displaystyle d=-1.} a 1 = a 6 − 5 d = 5. {\displaystyle a_{1}=a_{6}-5d=5.} a n = 5 + ( n − 1 ) × ( − 1 ) = 6 − n . {\displaystyle a_{n}=5+(n-1)\times (-1)=6-n.} a 2 = 4 , a 3 = 3 , a 4 = 2 , a 5 = 1. {\displaystyle a_{2}=4,a_{3}=3,a_{4}=2,a_{5}=1.} a 3 {\displaystyle a_{3}} b 1 = 4 , {\displaystyle b_{1}=4,} q = 1 2 {\displaystyle q={\frac {1}{2}}} b n = 4 × ( 1 2 ) n − 1 = 2 3 − n . {\displaystyle b_{n}=4\times \left({\frac {1}{2}}\right)^{n-1}=2^{3-n}.} T n = ( 6 − 1 ) × 2 3 − 1 + ( 5 − 1 ) × 2 3 − 2 + ( 4 − 1 ) × 2 3 − 3 + ⋅ ⋅ ⋅ + ( 6 − n ) × 2 3 − n . {\displaystyle T_{n}=(6-1)\times 2^{3-1}+(5-1)\times 2^{3-2}+(4-1)\times 2^{3-3}+\cdot \cdot \cdot +(6-n)\times 2^{3-n}.} 1 2 T n = ( 6 − 1 ) × 2 3 − 2 + ( 5 − 1 ) × 2 3 − 3 + ⋅ ⋅ ⋅ + ( 7 − n ) × 2 3 − n + ( 6 − n ) × 2 2 − n . {\displaystyle {\frac {1}{2}}T_{n}=(6-1)\times 2^{3-2}+(5-1)\times 2^{3-3}+\cdot \cdot \cdot +(7-n)\times 2^{3-n}+(6-n)\times 2^{2-n}.} { ( 1 A + B ) ( 1 2 ) 1 − B = 20 , ( 2 A + B ) ( 1 2 ) 2 − B = 28. {\displaystyle \color {BrickRed}{\begin{cases}(1A+B)\left({\frac {1}{2}}\right)^{1}-B=20,\\(2A+B)\left({\frac {1}{2}}\right)^{2}-B=28.\end{cases}}} { A = 8 , B = − 32. {\displaystyle \color {BrickRed}{\begin{cases}A=8,\\B=-32.\end{cases}}} T n = ( 8 n − 32 ) × ( 1 2 ) n + 32 = 8 n − 32 2 n + 32. {\displaystyle \color {BrickRed}T_{n}=(8n-32)\times \left({\frac {1}{2}}\right)^{n}+32={\frac {8n-32}{2^{n}}}+32.} 1 2 T n = 5 × 2 2 − ( 2 1 + 2 0 + ⋅ ⋅ ⋅ + 2 3 − n ) − ( 6 − n ) × 2 2 − n = 4 n − 16 2 n + 16. {\displaystyle {\frac {1}{2}}T_{n}=5\times 2^{2}-(2^{1}+2^{0}+\cdot \cdot \cdot +2^{3-n})-(6-n)\times 2^{2-n}={\frac {4n-16}{2^{n}}}+16.} T n = 8 n − 32 2 n + 32. {\displaystyle T_{n}={\frac {8n-32}{2^{n}}}+32.} 3. {\displaystyle 3.} x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1} F ( ± c , 0 ) {\displaystyle F(\pm c,0)} x {\displaystyle x} θ , {\displaystyle \theta ,} d = 2 a ⋅ a 2 − c 2 a 2 − c 2 cos 2 θ , {\displaystyle d=2a\cdot {\frac {a^{2}-c^{2}}{a^{2}-c^{2}\cos ^{2}\theta }},} θ = 90 ∘ {\displaystyle \theta =90^{\circ }} d 0 = 2 b 2 a . {\displaystyle d_{0}={\frac {2b^{2}}{a}}.} d ′ = 2 a ⋅ | a 2 − c 2 a 2 − c 2 cos 2 θ | . {\displaystyle d'=2a\cdot \left|{\frac {a^{2}-c^{2}}{a^{2}-c^{2}\cos ^{2}\theta }}\right|.} D ( x 0 , y 0 ) {\displaystyle D(x_{0},y_{0})} x 2 = 2 p y {\displaystyle x^{2}=2py} N {\displaystyle N} ( x 0 2 , 0 ) . {\displaystyle ({\frac {x_{0}}{2}},0).} D ′ ( x 0 ′ , y 0 ′ ) {\displaystyle D'(x_{0}',y_{0}')} y {\displaystyle y} N ′ {\displaystyle N'} ( 0 , y 0 ′ 2 ) . {\displaystyle (0,{\frac {y_{0}'}{2}}).} 2 {\displaystyle 2} 1. {\displaystyle 1.} 3 − 3 {\displaystyle 3-3} p V = n R T {\displaystyle pV=nRT} n {\displaystyle n} 2016 {\displaystyle 2016} I I {\displaystyle II} d 0 {\displaystyle d_{0}} m 1 , {\displaystyle m_{1},} m 2 . {\displaystyle m_{2}.} d 0 = m 1 m 2 . {\displaystyle d_{0}={\frac {m_{1}}{m_{2}}}.} p 1 V 1 n 1 = p 2 V 2 n 2 . {\displaystyle {\frac {p_{1}V_{1}}{n_{1}}}={\frac {p_{2}V_{2}}{n_{2}}}.} m 1 m 2 = n 1 n 2 . {\displaystyle {\frac {m_{1}}{m_{2}}}={\frac {n_{1}}{n_{2}}}.} d 0 = p 1 V 1 p 2 V 2 = 20 p 0 ⋅ 0.08 m 3 p 0 ⋅ 0.36 m 3 = 4.4 {\displaystyle d_{0}={\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {20p_{0}\cdot 0.08m^{3}}{p_{0}\cdot 0.36m^{3}}}=4.4} d ′ = 2 p 0 ⋅ 0.08 m 3 p 0 ⋅ 0.36 m 3 = 0.4 {\displaystyle d'={\frac {2p_{0}\cdot 0.08m^{3}}{p_{0}\cdot 0.36m^{3}}}=0.4} 4 {\displaystyle 4} ( i ) {\displaystyle (i)} ( i i ) {\displaystyle (ii)} T 0 = 300 K {\displaystyle T_{0}=300K} p 0 {\displaystyle p_{0}} T 1 = 303 K . {\displaystyle T_{1}=303K.} p 0 T 0 = p 1 T 1 . {\displaystyle {\frac {p_{0}}{T_{0}}}={\frac {p_{1}}{T_{1}}}.} p 1 = 1.01 p 0 . {\displaystyle p_{1}=1.01p_{0}.} m . {\displaystyle m.} ( p 1 − p 0 ) S = m g . {\displaystyle (p_{1}-p_{0})S=mg.} m g = 1 100 p 0 S . {\displaystyle mg={\frac {1}{100}}p_{0}S.} n 0 . {\displaystyle n_{0}.} n 1 , {\displaystyle n_{1},} n 0 T 0 = n 1 T 1 . {\displaystyle n_{0}T_{0}=n_{1}T_{1}.} n 1 n 0 = T 0 T 1 = 100 101 . {\displaystyle {\frac {n_{1}}{n_{0}}}={\frac {T_{0}}{T_{1}}}={\frac {100}{101}}.} p 2 . {\displaystyle p_{2}.} p 0 n 0 = p 2 n 1 . {\displaystyle {\frac {p_{0}}{n_{0}}}={\frac {p_{2}}{n_{1}}}.} p 2 = 100 101 p 0 . {\displaystyle p_{2}={\frac {100}{101}}p_{0}.} F = ( p 0 − p 2 ) S + m g = 1 101 p 0 S + 1 100 p 0 S = 201 10100 p 0 S . {\displaystyle F=(p_{0}-p_{2})S+mg={\frac {1}{101}}p_{0}S+{\frac {1}{100}}p_{0}S={\frac {201}{10100}}p_{0}S.} 2. {\displaystyle 2.} 3 − 5 {\displaystyle 3-5} v 0 {\displaystyle v_{0}} m 1 {\displaystyle m_{1}} m 2 {\displaystyle m_{2}} v 1 , {\displaystyle v_{1},} v 2 , {\displaystyle v_{2},} v 1 = m 1 − m 2 m 1 + m 2 ⋅ v 0 , v 2 = 2 m 1 m 1 + m 2 ⋅ v 0 . {\displaystyle v_{1}={\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}\cdot v_{0},v_{2}={\frac {2m_{1}}{m_{1}+m_{2}}}\cdot v_{0}.} v 2 − v 1 = v 0 {\displaystyle v_{2}-v_{1}=v_{0}} r , {\displaystyle r,} E p = k Q r . {\displaystyle E_{p}=k{\frac {Q}{r}}.} Q {\displaystyle Q} 4. {\displaystyle 4.} 1 k , {\displaystyle {\frac {1}{k}},} R {\displaystyle R} k 2 {\displaystyle k^{2}} I {\displaystyle I} 9 R {\displaystyle 9R} U = 220 V , {\displaystyle U=220V,} 22 V , {\displaystyle 22V,} 198 V , {\displaystyle 198V,} 66 V . {\displaystyle 66V.} A . {\displaystyle A.} k , {\displaystyle k,} 5 k 2 Ω {\displaystyle 5k^{2}\Omega } 4 I , {\displaystyle 4I,} 1 4 , {\displaystyle {\frac {1}{4}},} 3 + 5 k 2 = 4 × ( 3 + k 2 ) , {\displaystyle 3+5k^{2}=4\times (3+k^{2}),} k = 3 , {\displaystyle k=3,} B . {\displaystyle B.} R 2 , R 3 {\displaystyle R_{2},R_{3}} P , {\displaystyle P,} n 1 : n 3 {\displaystyle n_{1}:n_{3}} 4 : 1. {\displaystyle 4:1.} 16 R 3 {\displaystyle 16R_{3}} U , {\displaystyle U,} P = U 2 16 R , {\displaystyle P={\frac {U^{2}}{16R}},} P 1 = U 2 R = 16 P , {\displaystyle P_{1}={\frac {U^{2}}{R}}=16P,} R 1 {\displaystyle R_{1}} A , B . {\displaystyle A,B.} 16 R 3 {\displaystyle 16R_{3}} U ′ , {\displaystyle U',} R 2 , R 3 {\displaystyle R_{2},R_{3}} 8 U ′ . {\displaystyle 8U'.} P = ( 8 U ′ ) 2 16 R = 4 U ′ 2 R , {\displaystyle P={\frac {(8U')^{2}}{16R}}={\frac {4U'^{2}}{R}},} R 1 {\displaystyle R_{1}} P 1 = U ′ 2 R = 1 4 P , {\displaystyle P_{1}={\frac {U'^{2}}{R}}={\frac {1}{4}}P,} D . {\displaystyle D.} k 2 R 2 R ⋅ U = k 2 U 2 . {\displaystyle {\frac {k^{2}R}{2R}}\cdot U={\frac {k^{2}U}{2}}.} k 2 U 2 ⋅ 1 k = U , {\displaystyle {\frac {k^{2}U}{2}}\cdot {\frac {1}{k}}=U,} k = 2. {\displaystyle k=2.} 2 : 1. {\displaystyle 2:1.} U ′ = 2 U + k 2 U 2 = 4 U . {\displaystyle U'=2U+{\frac {k^{2}U}{2}}=4U.} C . {\displaystyle C.}
