ข้อสอบ A-Level คณิตประยุกต์ 1 ปี 2560 (วิชาสามัญเก่า)

ข้อ 25

กำหนดให้ straight a subscript straight n เป็นลำดับเลขคณิต ซึ่งมี a subscript 1 equals 2 และผลต่างร่วมเท่ากับ negative 2 over 9

ถ้า straight b subscript straight n equals 2 to the power of straight a subscript straight n end exponent แล้วจำนวนเต็มบวก straight m ที่น้อยที่สุดที่ทำให้  straight b subscript 1 times straight b subscript 2 times straight b subscript 3 times... times straight b subscript straight m greater or equal than 1024 เท่ากับข้อใดต่อไปนี้

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เฉลยข้อสอบ

begin mathsize 12px style น ิ ยาม space space space space space space a greater than 1 space space space ถ ้ า space a to the power of m greater than a to the power of n space space space space space space space จะได ้ space space space space m greater than n
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a to the power of m less than a to the power of n space space space space space space space จะได ้ space space space space m less than n end style

begin mathsize 12px style จากโจทย ์ space space space space space space space space space space space space space b subscript 1 times b subscript 2 times b subscript 3 times horizontal ellipsis times b subscript m space greater or equal than space 1024 space space space space space space semicolon space โดย space b subscript n equals 2 to the power of a subscript n end exponent
จะได ้ space space space space space space space space space space space space space space space space 2 to the power of a subscript 1 end exponent times 2 to the power of a subscript 2 end exponent times 2 to the power of a subscript 3 end exponent times horizontal ellipsis times 2 to the power of a subscript m end exponent space greater or equal than space 1024
space space space space space space space space space space space space space space space space space space space space space space space space space space space 2 to the power of a subscript 1 plus a subscript 2 plus a subscript 3 plus horizontal ellipsis plus a subscript m end exponent space greater or equal than space 2 to the power of 10
space space space space space space space space space space space space space space space space space space space space space thin space thin space thin space a subscript 1 plus a subscript 2 plus a subscript 3 plus horizontal ellipsis plus a subscript m space greater or equal than space 10 space rightwards arrow space box enclose 1

จากโจทย ์ space space space space space space space a subscript n space เป ็ นลำด ั บเลขคณ ิ ต space
thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space thin space ซ ึ่ งม ี space a subscript 1 equals 2 space และผลต ่ างร ่ วมเท ่ าก ั บ space minus 2 over 9
แสดงว ่ า space space space space space space space space space space a subscript 1 equals 2 space comma space d equals negative 2 over 9 end style

begin mathsize 12px style ส ู ตร space space space space space space space space space space space space space space S subscript n space equals space n over 2 open parentheses a subscript 1 plus a subscript n close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space n over 2 open square brackets 2 a subscript 1 plus open parentheses n minus 1 close parentheses d close square brackets

จาก space box enclose 1 space จะได ้ space space space space fraction numerator space m over denominator 2 end fraction open square brackets 2 a subscript 1 plus open parentheses m minus 1 close parentheses d close square brackets space greater or equal than space 10 end style

begin mathsize 12px style space space space space space space space space space space space space space space space space space space space m over 2 open square brackets 2 open parentheses 2 close parentheses plus open parentheses m minus 1 close parentheses open parentheses negative 2 over 9 close parentheses close square brackets space greater or equal than space 10
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space m open parentheses 4 minus 2 over 9 m plus 2 over 9 close parentheses space greater or equal than space 20
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 36 m minus 2 m squared plus 2 m space greater or equal than space 180
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 0 space greater or equal than space m squared minus 19 m plus 90
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space m squared minus 19 m plus 90 space less or equal than space 0
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses m minus 9 close parentheses open parentheses m minus 10 close parentheses space less or equal than space 0 end style

                 

begin mathsize 12px style space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets 9 comma 10 close square brackets

ด ั งน ั้ น space space space space space space space space space space space space จำนวนเต ็ มบวก space m space ท ี่ น ้ อยท ี่ ส ุ ด space เท ่ าก ั บ space 9 end style

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