Calculus Sequences and Series FAMAT State Convention For all questions, answer E. NOTA means none of the above answers are correct.

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1 Calculus Sequeces ad Series FAMAT State Covetio 005 For all questios, aswer meas oe of the above aswers are correct.. What should be the et logical term i the sequece, 5, 9,, 0? A. 6 B Give the ifiite sequece,,,,,,,,,, deote the first occurrece of ay umber special. For eample, terms,,, ad 7 above are special. How may terms from the 005 th term is the closest special term? A. B Evaluate. =0 A. B J.J. ad Luol play the followig game, which eds after somebody wis. J.J. rolls a regular 6-sided die. If he rolls a or a, he wis, otherwise Luol flips a coi, ad if he flips a head, he wis. If either J.J. or Luol has wo after this, the process is repeated util someoe emerges victorious. If J.J. ad Luol play this game,000 times, approimately how may times should Luol epect to wi? A. 85 B Evaluate 0 r 0 r r= 0 r () (). A B What is/are the solutio(s) to the followig equatio: (ifiitely) = A. ½ oly B. ½ ad 5 oly ± = page

2 Calculus Sequeces ad Series FAMAT State Covetio Evaluate A. 5 B. 5 E. NOTA 8. A ifiite amout of circles are draw, each with a radius half that of the circle precedig it. If the sum of the areas of these circles is equal to the sum of the circumfereces of these circles, what is the diameter of the largest circle? A. B Cosider the sequece a, with a = ad a =, where a = a (-) + a (-), for >. What is a 0 - a 8? A. 8 B Suppose f() = (-)f(+). What is f ( )? = A. 0 B Shelde buys a share of stock with his life savigs of $0. Each moth, the price of Shelde s stock goes up by %. After four years, Shelde sells the stock, ad wishes to buy as may shares of a ew stock, which costs $.5, as he ca. Sice Shelde oly has as much moey as he made from the stock sale, what is the maimum amout of whole shares of the ew stock that he ca buy? A. B. 5. For how may positive itegers are. < 00 ad >0000? A. B.. The umber of jelly beas Chris wats o day, give by J, ca be defied 9000 recursively as J + = J +. O day, Chris wats jelly bea. How may jelly J beas does Chris wat daily i the log ru? A. B page

3 Calculus Sequeces ad Series FAMAT State Covetio 005. Let F deote the th term of the Fiboacci sequece. As gets large, what is A. B Give the series a = +, at which term does a = 000? F? A. 000 B A ifiite arithmetic sequece with itegral differece has first term ad also a term which is 5. What is the sum of the possible values of the 5 th term of this series? A. 7 B What is the sum of the odd terms of a ifiite geometric sequece of real umbers with st term ad 5 th term? A. 8+ B. 8-6 Caot be determied F 8. The sum = ( + ) p+q? ca be epressed as a fractio i lowest terms, q p. What is A. 6 B For which of the followig does the ifiite series + = I. ( ) a = II. a = III. e a = IV. a = a coverge? A. I, III oly B. I, II oly III, IV oly I,III,IV oly 0. If = a, for some positive iteger a ad positive, how may of the followig caot be true of? a. is odd b. is prime c. is less tha a d. is ot a iteger A. B. page

4 Calculus Sequeces ad Series FAMAT State Covetio 005. What is the sum of all itegers betwee ad icludig 560 ad 56? A. B For 0 < <, which of the followig is equal to ( ( ) ) = 0 A. B. ( )( + ) ( )( ). Which of the followig is equal to 0! ?!! 5! 8! 9! A. si() + cos() + e B. si() - cos() + e si() - cos() + e si(cos(e - )). Which of the followig is equal to the sum of the real coefficiets of (+i)? (i = the square root of ) A. - π si( ) B. / π cos( ) - π cos( ) / π si( ) 5. Lee had 5 dollars i his piggy bak this morig. Every afteroo, his mother matches the amout he has i his piggy bak (i.e. doubles the amout he has i his piggy bak), ad every ight Lee smuggles out dollars to go buy pecils. If Lee has $ i his piggy bak ights from toight, before he smuggles out his dollars for the ight, what is (to the earest hudredth)? A. $.00 B. $.67 $.00 $ Which of the followig is the largest (aswer NOTA if there is a tie)? A. The largest -digit Fiboacci umber B. The largest -digit triagular umber The largest -digit prime umber The largest -digit square umber page

5 Calculus Sequeces ad Series FAMAT State Covetio Joh got A s, B s, ad C s o his report card, ad is ot sure i which order to read these letter grades (ad oly the letters) to his parets. He kows that as log as he starts off with a A ad eds with a A, his parets will forget about his lesser grades. How may differet ways ca he read off his grades while makig his parets happy? A. 0 B Let the sequece A have terms a =, ad the sequece B have terms b = 80-. For what do A ad B have the same value? A. 8 B Mike drops a ball from the top of a 50 foot buildig. Each time the ball bouces, it rebouds to / of its previous height. How far does the ball travel i total (i feet)? A. 00 B Let the sequece A have terms a =, ad let the sequece B have terms b = a i. Which of the followig are true about B? i= I. Its terms are all square umbers II. Its terms are all odd umbers III. Every b is divisible by IV. There eists a b such that b is less tha two times a A. I oly B. I, III, ad IV oly II ad III oly I ad IV oly. page 5