2011 Λύσεις Σχ. β&I

3 i œûùˆ ìè áâˆìâùúèî úfiôô ìâ

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3. i) ŒÛÙˆ ÌÈ· ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ Ì ÚÒÙÔ fiÚÔ · 1 Î·È ÏfiÁÔ Ï. ΔfiÙ ÔÈ fiÚÔÈ Ù˘ ÚÔfi‰Ô˘ ›ӷÈ: · 1 , · 1 Ï, · 1 Ï 2 , · 1 Ï 3 , ..., · 1 Ï Ó , ... Î·È Ù· ÙÂÙÚ¿ÁˆÓ· ÙˆÓ fiÚˆÓ ·˘ÙÒÓ Â›Ó·È: · 1 2 , · 1 2 Ï 2 , · 1 2 Ï 4 , · 1 2 Ï 6 , ... · 1 2 Ï . ¶·Ú·ÙËÚԇ̠fiÙÈ Ë ·ÎÔÏÔ˘ı›· ·˘Ù‹ Â›Ó·È ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ Ì 1Ô fiÚÔ · 1 2 Î·È ÏfiÁÔ Ï 2 . ii) ∞Ó ˘„ÒÛÔ˘Ì ÙÔ˘˜ fiÚÔ˘˜ Ù˘ ÚÔfi‰Ô˘ ÛÙËÓ k ¤¯Ô˘ÌÂ: · 1 k , · 1 k Ï k , · 1 k Ï 2k , · 1 k Ï 3k , ... · 1 k Ï Ók ¶·Ú·ÙËÚԇ̠fiÙÈ Ë ·ÎÔÏÔ˘ı›· ·˘Ù‹ Â›Ó·È ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ Ì 1Ô fiÚÔ · 1 k Î·È ÏfiÁÔ Ï k . 4. i) Œ¯Ô˘Ì √È (1) Î·È (2) Û¯ËÌ·Ù›˙Ô˘Ó ÙÔ Û‡ÛÙËÌ· · 1 + · 1 Ï = 3 + 3 (1) Î·È · 1 Ï 4 – 1 Ï – 1 = 4(3 + 3) (2). Ó = 13 ± 225 2 = 13 ± 15 2 = 10Ó + 4 = (Ó – 5)(Ó + 2) ( 10Ó + 4 4 ) 2 = Ó – 5 Ó + 2 Ï = 2 3 Î·È · 1 = 2 9 . · Ó + 1 · Ó = 2 Ó + 1 3 Ó + 2 2 Ó 3 Ó + 1 = 2 Ó + 1 3 Ó + 1 2 Ó 3 Ó + 2 = 2 3 · Ó + 1 = · Ó 2 3 . 1 3 4 = 60 81 = 20 27 0,74m. 1 3 3 ∫∂º∞§∞π√ 5: ¶ƒ√√¢√π 74
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· 1 (Ï + 1) = 3 + · 1 3 + Ï 2 + Ï + 1) = 4(3 + ). ªÂ ‰È·›ÚÂÛË Î·Ù¿ ̤ÏË ÙˆÓ ÂÍÈÛÒÛÂˆÓ ÙÔ˘ Û˘ÛÙ‹Ì·ÙÔ˜ ÚÔ·ÙÂÈ Ï 3 + Ï 2 – 3Ï – 3 = 0 Ï 2 (Ï + 1) –3(Ï + 1) = 0 (Ï + 1)(Ï 2 – 3) = 0 Ï = –1 Ï = Ï = ∞ÓÙÈηıÈÛÙԇ̠ÙȘ ÙÈ̤˜ ·˘Ù¤˜ ÙÔ˘ Ï ÛÙËÓ (1) Î·È ¤¯Ô˘Ì °È· Ï = –1, · 1 . 0 = 3 + (·‰‡Ó·ÙÔ) °È· Ï = , · 1 ( + 1) = 3 + · 1 = °È· Ï = , · 1 (1 – ) = 3 + · 1 = –(3 + 2 ) . 5. Œ¯Ô˘Ì · 1 Ï + · 1 Ï 5 = 34 · 1 Ï(Ï 4 + 1) = 34 (1) Î·È · 1 Ï 2 + · 1 Ï 6 = 68 · 1 Ï 2 4 + 1) = 68 (2) ªÂ ‰È·›ÚÂÛË Î·Ù¿ ̤ÏË ÙˆÓ (1) Î·È (2) ¤¯Ô˘ÌÂ Ï = 2, ÔfiÙ Ì ·ÓÙÈηٿ- ÛÙ·ÛË ÛÙËÓ (1) ‚Ú›ÛÎÔ˘Ì · 1 = 1. ÕÚ· S 10 = 1 . = 1024 – 1 = 1023 . 6. ∞Ó · Ó Â›Ó·È Ô ÏËı˘ÛÌfi˜ Ù˘ ¯ÒÚ·˜ ‡ÛÙÂÚ· ·fi Ó ¯ÚfiÓÈ· ·fi Û‹ÌÂÚ·, ÙfiÙ ÙÔÓ ÂfiÌÂÓÔ ¯ÚfiÓÔ, ‰ËÏ·‰‹ ‡ÛÙÂÚ· ·fi Ó + 1 ¯ÚfiÓÈ· ·fi Û‹ÌÂÚ·, ı· Â›Ó·È (Û ÂηÙÔÌ̇ÚÈ·). · Ó + 1 = · Ó + . · Ó = 1,02 . · Ó . ÕÚ· Ô ·Ó·‰ÚÔÌÈÎfi˜ Ù‡Ô˜ Ù˘ ·ÎÔÏÔ˘ı›·˜ Â›Ó·È · Ó + 1 = 1,02 . · Ó . ∂Âȉ‹ · 1 = 90 . 1,02 Î·È · Ó + 1 = 1,02 . · Ó Ë ·ÎÔÏÔ˘ı›· Â›Ó·È ÁˆÌÂÙÚÈ΋ ÚfiÔ‰Ô˜ Ì 1Ô fiÚÔ · 1 = 90 . 1,02 Î·È ÏfiÁÔ Ï = 1,02, ÂÔ̤ӈ˜ · Ó = 90 . 1,02 . 1,02 Ó – 1 · Ó = 90 . 1,02 Ó . ⁄ÛÙÂÚ· ·fi 10 ¯ÚfiÓÈ· Ô Ï˘ı˘ÛÌfi˜ Ù˘ ¯ÒÚ·˜ ı· Â›Ó·È · 10 = 90 .
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