Otsuma Nakano Senior High School Spring Seminar Mathematics B

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1 Otsuma Nakano Senior High School Spring Seminar Mathematics B

2 2 a d a n = a + (n 1)d 1 2 ( ) {( ) + ( )} = n 2 {2a + (n 1)d} a r a n = ar n 1 a { r ( ) 1 } r 1 = a { 1 r ( )} 1 r (r 1) n 1 = n k=1 n k = 1 n(n + 1) 2 k=1 n k 2 = 1 n(n + 1)(2n + 1) 6 k=1 n { } k 3 = n(n + 1) k=1 n a k k=1 1 (pn + a)(pn + b) b n = a n+1 a n n 2 a n = a 1 + n 1 b k k=1 n 2 a n = S n S n 1 n = 1 a 1 = S 1

3 3 1 a n+1 = a n + d a 1 = a a n = a + (n 1)d 2 a n+1 = ra n a 1 = a a n = ar n 1 113(1)(2) {a n } (1) a 1 = 3, a n+1 = a n + 4 (2) a 1 = 4, 2a n+1 + 3a n = 0

4 4 3 a n+1 = a n + f(n) a n+1 a n = f(n) {a n } {f(n)} n 2 a n = a 1 + n 1 k=1 f(k) 113(3) {a n } (3) a 1 = 1, a n+1 = a n + 2 n 3n + 1

5 5 119 a 1 = 2, a n+1 = n + 2 n a n + 1 {a n } (1) a n n(n + 1) = b n b n+1 b n n (2) a n n

6 6 4 a n+1 = pa n + q 2 c = pc + q c a n+1 c = p(a n c) b n = a n c b n+1 = pb n {a n } a 1 = 6, a n+1 = 2a n 3

7 7 5 a n+1 = pa n + qr n p n+1 a n+1 p n+1 b n+1 = b n + qrn p n+1 r n+1 a n+1 r n+1 = a n p n + qrn p n+1 3 = p r an r n b n = a n p n + q r b n = a n r n b n+1 = p r b n + q r a 1 = 3, a n+1 = 2a n + 3 n+1 {a n }

8 8 6 a n+1 = pa n ra n + s a n 0 a n a n+1 = ra n + s pa n = r p + s p 1 a n b n = 1 a n b n+1 = r p + s p b n a 1 = 1 5, a n+1 = a n 4a n 1 {a n}

9 9 7 a n+1 = pa n q (p > 0) a n > 0 a n+1 > 0 p log p a n+1 = log p pa n q = 1 + q log p a n b n = log p a n b n+1 = 1 + qb n 4 b n = log p a n a n = p b n 118 a 1 = 1, a n+1 = 2 a n {a n }

10 10 8 S n a 1 = S 1, S n+1 S n = a n+1 S n {a n } n S n a n S n = 2a n 2n + 5 a n n

11 11 9 a n+1 = f(n)a n a n = f(n 1) f(n 2) f(n 3) f(2) f(1)a a 1 = 1 2, (n + 1)a n = (n 1)a n 1 (n 2) {a n }

12 12 10 a n+1 = pa n + qn + r b n = a n αn β b n+1 = pb n α, β 2 α, β b n = a n αn β b n+1 = pb n a n+1 α(n + 1) β = p(a n αn β) a n+1 = pa n + (α pα)n + α + β pβ (α pα)n + α + β pβ = qn + r α, β n n + 1 a n+2 = pa n+1 + q(n + 1) + r ) a n+1 = pa n + qn + r a n+2 a n+1 = p(a n+1 a n ) + q b n = a n+1 a n b n+1 = pb n + q 4 a n+1 a n = b n a 1 = 1, a n+1 = 3a n + 4n {a n }

13 13

14 14 11 a n+2 = pa n+1 + qa n p + q = 1 a n+1 a n+2 a n+1 = (p 1)a n+1 (p 1)a n = (p 1)(a n+1 a n ) b n = a n+1 a n b n+1 = (p 1)b n 2 {b n } a n+1 a n = b n 3 p + q 1 t 2 = pt + q α, β a n+2 αa n+1 = β(a n+1 αa n ) a n+2 βa n+1 = α(a n+1 βa n ) c n = a n+1 αa n, d n = a n+1 βa n c n+1 = βc n, d n+1 = αd n 2 α β c n d n = ( α + β)a n a n α = β 5

15 15 121(1) {a n } (1) a 1 = 1, a 2 = 2, a n+2 + 4a n+1 5a n = 0

16 16 121(2) {a n } (2) a 1 = 0, a 2 = 1, a n+2 = a n+1 + 6a n

17 {a n } a 1 = 0, a 2 = 2, a n+2 4a n+1 + 4a n = 0

18 18 12 a n+1 = pa n + qb n, b n+1 = ra n + sb n a n+1 + αb n+1 = k(a n + αb n ) k, α c n = a n + αb n c n+1 = kc n {a n }, {b n } a 1 = b 1 = 1, a n+1 = a n + 4b n, b n+1 = a n + b n (1) a n+1 + xb n+1 = y(a n + xb n ) x, y 2 (2) {a n }, {b n }

19 {a n }, {b n } a 1 = 1, b 1 = 1, a n+1 = 5a n 4b n, b n+1 = a n + b n (1) a n+1 + xb n+1 = y(a n + xb n ) x, y (2) {a n }, {b n }

20 20 13 a n+1 = pa n + q ra n + s x = px + q rx + s 125 a 1 = 1, a n+1 = a n 9 a n 5 {a n} (1) n a n 3 (2) b n = 1 a n 3 b n+1 b n a n

21 {a n } a 1 = 4, a n+1 = 4a n + 8 a n + 6 (1) b n = a n β a n α b n α, β(α > β) (2) a n

22 a 1 = 1, a n+1 = (1) a 2, a 3, a 4 a n 1 + 3a n {a n } (2) a n n

23 {a n } a n > 0 a n (a 1 + a 2 + a n ) 2 = a a a n 3

24

(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n

(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n . 99 () 0 0 0 () 0 00 0 350 300 () 5 0 () 3 {a n } a + a 4 + a 6 + + a 40 30 53 47 77 95 30 83 4 n S n S n = n = S n 303 9 k d 9 45 k =, d = 99 a d n a n d n a n = a + (n )d a n a n S n S n = n(a + a n

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