2011 Λύσεις Σχ. β&I

Fi ùô ùô ó 1 ó 1ˆ ôìâ 195 5 ó 1 5 ó 39

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∞fi ÙÔ Ù‡Ô · Ó = · 1 + (Ó – 1)ˆ ¤¯Ô˘Ì 195 = 5 + (Ó – 1) . 5 Ó = 39. ∂Ô̤ӈ˜ S 39 = (5 + 195) = 39 . 100 = 39000 . ii) ΔÔ ˙ËÙÔ‡ÌÂÓÔ ¿ıÚÔÈÛÌ· Â›Ó·È ÙÔ 12 + 15 + ... + 198 Î·È ÔÈ ÚÔÛıÂÙ¤ÔÈ ÙÔ˘ Â›Ó·È ‰È·‰Ô¯ÈÎÔ› fiÚÔÈ ·ÚÈıÌËÙÈ΋˜ ÚÔfi‰Ô˘ Ì · 1 = 12, ˆ = 3 Î·È · Ó = 198. ∞fi ÙÔ Ù‡Ô · Ó = · 1 + (Ó – 1)ˆ ¤¯Ô˘Ì 198 = 12 + (Ó – 1) . 3 ‹ Ó = 63. ∂Ô̤ӈ˜ S 63 = (12 + 198) = . 210 = 63 . 105 = 6615 . 4. i) Œ¯Ô˘Ì · Ó + 1 – · Ó = 5(Ó + 1) –4 – 5Ó + 4 = 5 ‹ · Ó + 1 = · Ó + 5. ∂Ô̤ӈ˜ Ë ·ÎÔÏÔ˘ı›· Â›Ó·È ·ÚÈıÌËÙÈ΋ ÚfiÔ‰Ô˜ Ì · 1 = 5 . 1 – 4 = 1, ˆ = 5 Î·È · 30 = 5 . 30 – 4 = 146, ÔfiÙ S 30 = (1 + 146) = 15 . 147 = 2205 . ii) Œ¯Ô˘Ì · Ó + 1 – · Ó = –5(Ó + 1) –3 + 5Ó + 3 ‹ · Ó + 1 = · Ó – 5. ∂Ô̤ӈ˜ Ë ·ÎÔÏÔ˘ı›· Â›Ó·È ·ÚÈıÌËÙÈ΋ ÚfiÔ‰Ô˜ Ì · 1 = –5 . 1 – 3 = –8, ˆ = –5 Î·È · 40 = –5 . 40 – 3 = –203, ÔfiÙ S 40 = (–8 – 203) = 20 . (–201) = –4220 . 5. ¶Ú¤ÂÈ ·fi ÙÔ ¿ıÚÔÈÛÌ· 1 + 2 + 3 + ... + 200 Ó· ·Ê·ÈÚ¤ÛÔ˘Ì ÙÔ ¿ıÚÔÈÛÌ· 4 + 8 + 12 + ... + 200 ÙˆÓ ÔÏÏ·Ï·Û›ˆÓ ÙÔ˘ 4 Î·È ÙÔ ¿ıÚÔÈÛÌ· 9 + 18 + 27 + ... + 198 ÙˆÓ ÔÏÏ·Ï·Û›ˆÓ ÙÔ˘ 9. ŸÌˆ˜ ÛÙ· ÔÏÏ·Ï¿ÛÈ· ÙÔ˘ 4 Î·È ÙÔ˘ 9 ÂÚȤ¯ÔÓÙ·È Î·È Ù· ÔÏÏ·Ï¿ÛÈ· ÙÔ˘ 36 Ô˘, Ì ·˘ÙfiÓ ÙÔÓ ÙÚfiÔ, ·Ê·ÈÚÔ‡ÓÙ·È ‰˘Ô ÊÔÚ¤˜. ¶Ú¤ÂÈ ÏÔÈfiÓ Ó· ÚÔÛı¤ÛÔ˘Ì ÌÈ· ÊÔÚ¿ Ù· ÔÏÏ·Ï¿ÛÈ· ÙÔ˘ 36 ÁÈ· Ó· ‚Úԇ̠ÙÔ Ú·ÁÌ·- ÙÈÎfi ¿ıÚÔÈÛÌ·. ∂Ô̤ӈ˜ S = (1 + 2 + 3 + ... + 200)(4 + 8 + 12 + ... + 200) – (9 + 18 + 27 + ... + 198 + + (36 + 72 + ... + 180). 40 2 30 2 63 2 63 2 39 2 182 2 ∫∂º∞§∞π√ 5: ¶ƒ√√¢√π 68
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∫·Ù¿ Ù· ÁÓˆÛÙ¿ ¤¯Ô˘ÌÂ: 1 + 2 + 3 + ... + 200 = (1 + 200) = 100 . 201 = 20100 4 + 8 + 12 + ... + 200 = (4 + 200) = 25 . 204 = 5100 9 + 18 + ... + 198 = (9 + 198) = 11 . 207 = 2277 36 + 72 + ... + 180 = (36 + 180) = . 216 = 5 . 108 = 540 ÕÚ· S = 20100 – 5100 – 2277 + 540 = 13263 . 6. ΔÔ ¿ıÚÔÈÛÌ· Ó fiÚˆÓ Ù˘ ·ÎÔÏÔ˘ı›·˜ Â›Ó·È S Ó = [2· 1 + (Ó – 1)ˆ] S Ó = [2 . 1 + (Ó – 1) . 2]. ¶Ú¤ÂÈ S Ó > 400 [2 . 1 + (Ó – 1) . 2] > 400 Ó 2 > 400 Ó > 20. 7. °È· ÙËÓ 1Ë ÁÚ·ÌÌ‹ ÙÔ˘ ›Ó·Î· ¤¯Ô˘ÌÂ: · Ó = · 1 + (Ó – 1)ˆ = 120 + (12 – 1)(–10) = 120 – 110 = 10 . S Ó = 1 + · Ó ) = (120 + 10) = 6 . 130 = 780 . °È· ÙËÓ 2Ë ÁÚ·ÌÌ‹ ¤¯Ô˘ÌÂ: · Ó = · 1 + (Ó – 1)ˆ 109 = 5 + (27 – 1)ˆ ˆ = 4 . S Ó = 1 + · Ó ) = (5 + 109) = . 114 = 1539 . °È· ÙËÓ 3Ë ÁÚ·ÌÌ‹ ¤¯Ô˘ÌÂ: S Ó = [2· 1 + (Ó – 1)ˆ] 210 = [2· 1 + 11 . 3] · 1 = 1 . · Ó = · 1 + (Ó – 1)ˆ · Ó = 1 + 11 . 3 · Ó = 34 . °È· ÙËÓ 4Ë ÁÚ·ÌÌ‹ ¤¯Ô˘ÌÂ: · Ó = · 1 + (Ó – 1)ˆ – 8 = · 1 + 15 . 2 · 1 = –38 . S Ó = 1 + · Ó ) S Ó = (–38 – 8) = 8 . (–46) = –368 . 16 2 Ó 2 12 2 Ó 2 27 2 27 2 Ó 2 12 2 Ó 2 Ó 2 Ó 2 Ó 2 5 2 5 2 22 2 50 2 200 2 5.2. ∞ÚÈıÌËÙÈ΋ ÚfiÔ‰Ô˜ 69
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8. ΔȘ ÚÒÙ˜ 12 ÒÚ˜ ÙÔ Ï‹ıÔ˜ ÙˆÓ ÎÙ‡ˆÓ Â›Ó·È 1 + 2 + 3 + ... + 12 = (1 + 12) = 6 .
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