(Sequence) (pattern) + + + +... + 9 + 98 + 99 + 00 =? 9 5050 00 00 + 00 = + 99 = + 98 = + 9 =... = 9 + 5 = 50 + 5 = 0 50 0 50 = 5050 00 ( + 00) + + +... + 98 + 99 + 00 = = 5050 Gauss, Carl Friedrich (-855) 99 9
+ + + +... + 9 + 98 + 99 + 00... S S(Sum of sequence) S () S () () 50 ( + 00) 50 S = = 550 () 50 ( + 99) 50 S = = 500 () = + 98 = 98 + = ( + 98) S = (**) = 650 = ( ) +
05 06 99 + =995 (**) (995 06) = + = 8 ( 06 + 995) 8 06 + + 0 +... + 995 = = 06,, 9,, 8,... 9 8 = = = =... = 9 (), 6,,, 8,... (),,,,,... 8 6 () -5, 5, -, 5, 5,... () 6 8 = = = =... = 6 () = 8 = 6 = =... = 8 6 () 5 5 = = 5 = 5 = 5 5 5,,,, 5, 8,,,, 55,...,,,, 5, 8,,,, 55,... 0 5 8... (Fibonacci, Leonardo)
00 00 00 5050 + + +...+ 98 +99 + 00 = 5050,,,..., 98, 99,00,,,..., 99,00,0 - = 0-00 = 00-99 = = 00 00 00 9900 0099 5050 5050 + 00 5050 + 00 99 5050 + 9800 00 5050 + 9900 =00(5050) + (00 + 00 + 00 +... + 9800 + 9900) = 505000 + 00( + + +... + 98 + 99) = 505000 + 95000 = 000000 (Natural Number)
5 6 8 9 0 = = + 6 = + + 0 = + + + = + + +...+ 8 + 9 +0 = 0 = 0+ =... () 50 () 5... () 50 9 50 ( + 9) 9 + + +... + 9 = = 5 5
5 + = 50 () + + +... + 5 = 0 5 0 5 5 6 5 6 00, 6,,,, 8,, 5, 8,,,,,, 9, 0, 5, 6, (a) (a) (b) (c) (d) 8 6 6,, 8, 6 = 6
(b) 6 006 98 8, 9, 9, 98, 99 88, 9, 9, 9, 00 89, 90, 95, 96, 00 00 + = 6 6 (b) = 8... 9 + 5 + + +... + 9 + 00 + + + + ( + 9) ( 5 + 9) = + + 00 = 66 8 (c) C, 0, 6, 80 + = 6 8 60. Natural Number 0 8 9 0 9 6
8 5 6 5. 00,99,98,9,99,98,9,96,98,9,96,95,... 9.,, 5, 5 5 5 9 9 9 5 989. 5 6 8 9 0 5 6 0 5. 6 5 5 8 9 0 8
99 6. 5 5 0 6 8 9 8 9 6 5 0 5 5 (a) 5 5 (b) 0. 5 9 59 (a) 0 (b) 0 (c) 0 8.... (a) 0 (b) 99 9.,,,,,,,,,,... (a) 0 (b) 00 (c) 00 9
0
0.,,,,,,,,,, 5,,, 5, 6, 5,... (a) 0 (b) 0 (c) 00 (d) 00. 5 6 8 9 0,,,,,, (a) 5 (b) 5 (c) 00 (d) 0.,, 5,,,, 5,,,,, 5,... (a) 5 (b) (c) (d)., 9, 9,,,,,, 9, 9,,,,,, 9, 9,,,,, 998 (a) (b) 9 (c) (d)
. 5 5 0 6 9 0 = 0 () 6 = + [( + (+++)] 6 = 0 = + [ + ( + + + +... + 9 + 9 + 0 + 0)] =. 0099989 00 999998996 9 9 = 8... 9 9 = 00 8 = 5
. 989 989 + Θ = 995 989 995 8 995 8 =... 99 8 98 989 98 98 99 9 985 98 989 99 989. 0 Θ Θ = 96 < 0 < 5 = 5 0 5 5 = 96 + = 9 0 9 + = 0 0 5 5. Θ 6 5 5 8 9 0 + + + + 6 = 95 < 99 + + + + 6 = 06 > 99 99 6 06 99 99 Θ 06 99 = 99 6 = 0
6. (a) 965 (b) 5 = 5 = 5 5 5 = 5 5 + = 5 6 5 0 6 5 9 = = + + ( ) 5 = + + ( ) 0 = + 9 + ( ) 5 = + 6 + (5 ) 6 6 = + 5 + (6 ) 0= + (0 ). (a) 0 0 0 + + + 0 = 55 (b) = () 5 = ( + ) = ( + + ) 9 = ( + + + ) ( ) 9 = ( + + + + 9) = (5) = 89 0 = 89 + = 9
0 = 9 + (0 ) = 09 0 (c) 0 = (9+ 09) = 000 8. 0... (a) 6 0 = + + + + 6 + 0 = 6 (b) (a) 99 Θ + + + + 6 = 95 < 99 + + + + 6 = 06 > 99 99 6 99 95 + = 99 9. (a) 0 = + + + + (b) (c) = 0 00 00 Θ + + + + = 9 < 00 + + + + = 05 > 00 00 00 = + + +... + + 9 = 95 0. (a) 5 6 + + + + 5 + 6 = 6 5
5,,, 5 (b) 5 ( + ) Θ + + + + = = 05 06 5 5 ( ) + (c) Θ + + + + = = 9 < 00 9 05 (d) 8 90 8 8 0 = +... + + + + + + 6 0 5 8 = + + + + + + + 5 6 8 = 8 + +... + + 8 + 8. 5,,,,,,,,,, 5,,, 5, 6, 5, (a) 0 8, 9, 0, 9, 8 5 0 8 (b) 0 5 (c) 00 0,,,, 0 00 0 (d) 9,, 9,, 5 = 0 + + + + 0 = 0 0 ( 9 + 0) 00 = = 0 6
.,, 5,,, 5,,,, 5 + + + + = 8 (a) 5 8 (b) (a) = 8 = 6 (c) (a)(b)5 8 + 8 + 0 = 5 (d) = 8 + 5 + 0 =., 9, 9,,,,,, 9, 9,,,,,, 9, 9,,,,, 998 = 85 85 85 86, 9, 9,,,,, 9, 9, 9 998 (a) 85 + = 856 (b) 9 85 + = 5 (c) 50 85 = (d) = 856 + 9 5 + 50 = 88