Mu Alpha Theta National Convention: Denver, 2001 Sequences & Series Topic Test Alpha Division

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Todd Banks
6 years ago
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1 Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio. Wht is the commo rtio of the geometric sequece, 7, 9,? 7 (C) 5. The commo differece of the rithmetic sequece,, 0, is 5 (C) 8. Fid the sum of the first eve turl umbers. 0 5 (C) 75. Wht is the sum of the first 8 odd positive itegers? 57 5 (C) Evlute: (C) 0. Ctl Numbers re defied explicitly by C = (with > 0) d proves to be + useful i solvig certi coutig problems. Fid the product of the first three Ctl Numbers. (C) Evlute: ! 7 (C) A sequece is defied explicitly by = + 5( ). Wht is the vlue of +? 7 0 (C) 0 70 Pge

2 Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio 9. A storeower is settig up impressive disply of spich cs to help with sles. The cs re to be plced o stc with 0 levels, ech level hvig oe more c th the oe bove it. Hlf of the cs i stoc will be used i his disply, with cs o the top level. How my cs does he hve i stoc? 55 5 (C) Evlute, where is perfect squre. = (C) ( + )(+ ) ( + )( +) + ( ) ( + )( + ). Two sequeces re defied s + = d b =. For wht vlues of is > b? 5 (C) > 5. If V =, evlute V. = = 5 95 (C) To help ese the cost of his prescriptio drug pymets, Wye decides to joi Drug Club, which helps its members sve moey o their medictio. Members must py $0 for their first moth d tht fee goes up $ ech moth. If Wye styed i the club for moths, how much moey did he py o dues loe? $09 $7 (C) $99 $8. Let T be the th trigulr umber (ssume is the first) d S the th squre umber (ssume is the first). Which of the followig is equl to? S T + T (C) T ( T ) Pge

3 Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio 5. How my umbers must be iserted betwee d 00 to me rithmetic sequece with commo differece? 8 9 (C) 0. Oce upo time, there were seve forests ech housig seve owls. Ech owl illed seve mice. If left live, ech mouse would hve ete seve ers of cor. Whe ot ete, ech er of cor produced seve pouds of gri. How my pouds of gri were sved due to the existece of the forests? 0 (C) Which of the followig is possible ordered triplet (, b, c) if 7 b 0 i 7 =? i= 0 c (,, ) (00, -, ) (C) (0, -, 8) (00,, 8) 8. Fid the sum of the first terms of the rithmetic series ( x+ ) + ( y) + ( x+ y ) (C) A rubber bll dropped from height of 0 meters rebouded o ech bouce 5 8 of the height from which it fell. How fr (i meters) did it trvel before comig to rest? 0 0 (C) Solve for x: ( i 8) = 0. 5 i= x 7 8 (C) Fid the mtrix equivlet to + + +! (C) Pge

4 Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio. The fifth term of rithmetic sequece is d the xth term is, where x > 5. Give tht the commo differece of this sequece is iteger, how my possible vlues re there for x? 5 8 (C) 0. A cube is iscribed i sphere of rdius 9. Aother sphere is iscribed iside the cube d secod cube is iscribed i this sphere. If this ptter cotiues, wht is the surfce re of the th sphere? π 8 π 79 (C) π π 87. Fid the sum of ll positive 7-digit plidromes. Express your swer i scietific ottio (C) [ + + ] 5. Evlute the ifiite series log (+ ) log (+ ). = l log l 5 (C) lo g log 8 Diverges 5 7 ( ). Evlute (( ) ( ) ) + +. = 0 0 (C) - 7. Strtig with circle of rdius, ew circle is formed whose rdius i uits is umericlly equl to the re of the previous circle i squre uits. Wht is the re of the th circle? 0 π π (C) π π 8. Let i =. For y positive iteger greter th, which of the followig is equl to = 0 i e π? (C) 0 - Pge

5 Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio 9. If is odd d is rithmetic sequece with positive terms, fid the mximum vlue of (C) Evlute log N, where x represets the gretest iteger less th or equl to x. N = 90 8 (C) Wht is the hrmoic me of the ltitudes of trigle whose re is umericlly equl to its perimeter? Assume cosistet uits (i.e. legth is uits, re is squre uits). 9 (C) 999. Express j( j+ )( j+ ) i terms of biomil coefficiets. j= r (C) For which vlue of v does the equlity true? (Assume >.) (... )... v v v v = hold No vlue (C). Let d b be two rithmetic progressios with > 0, the sum of the first terms of which re ( ) d S ( ), respectively. Give tht S b S ( ) = S ( ) + 8, fid 5 b. b (C) 5 9 Pge 5

6 Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio 5. A recursive sequece is defied by = + =. Wht s the remider whe is divided by 0? 9 where =, = 5, d 0 7 (C) 5. Ry d Bri decide to ply gme. At the strt of ech tur, the plyer rolls fir - sided die. If prime umber is rolled, tht plyer loses (d the other plyer is declred the wier). If is rolled, tht plyer wis. If either of these evets occurs, the ext plyer s tur begis. The gme cotiues util someoe wis. Ry decides to go first. Wht is the probbility tht sigle gme goes o idefiitely? / / (C) / / 7. Which of the followig is equl to x + x + x + x + 5x + x + 5x + x + x + x + x for x <? x ( x) ( x ) i x (C) i= 0 x x x ( x ) ( x) 8. Give tht = π = d = ( ) π = 8, fid the sum = +. π π (C) π π 5 9. Evlute (C) 0. Wht is the iteger prt of the sum ? (C) 5 Pge

EXERCISE a a a 5. + a 15 NEETIIT.COM

EXERCISE a a a 5. + a 15 NEETIIT.COM - Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()