Alpha Sequences and Series FAMAT State Convention 2017

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1 Alpha Sequences and Series FAMAT State Convention 017 For all questions, E NOTA means none of the above answers is correct. 1. We say that a number is arithmetically sequenced if the digits, in order, form an arithmetic sequence. Compute the number of 4-digit positive integers (none beginning with 0) which are arithmetically sequenced. 30 b) c) 1 d) 13. A ball is dropped from a height of 36 meters. Each time it strikes the ground it rebounds to a height of /3 of the distance from which it fell. Find the total distance (in meters) traveled by the ball before it comes to rest. 108 b) 144 c) 180 d) You are walking up a staircase with stairs (of negligible width) that are 1ft. tall and 1ft. deep. The number of stairs from the ground floor to the first floor is 1. From the first floor to the second is stairs. From the 99th to the hundredth stair is 100 steps. At each floor, the staircase makes a 90 turn to the right. At the top of the 100th floor, how far away (in feet) from the bottom of the staircase are you? b) c) d) A sequence is generated as follows: if the term; otherwise it is two more than twice the term? th n term is even, then the n 1 th term is half the n th th n term. If the first term is 10, what is the 008 th 5 b) 6 c) 8 d) 1 5. Consider the sequence 1,, 1,,, 1,,,, 1,,,,, 1,... Find n such that the first n terms sum up to b) 107 c) 118 d) 1356 Page 1 of 6

2 Alpha Sequences and Series FAMAT State Convention Evaluate: b) c) d) The measures of the 7 angles of a convex heptagon are in arithmetic progression. If the measure of the smallest angle is 101, then the largest angle, to the nearest degree, has what measure? 108 b) 147 c) 156 d) The n th term of a sequence is given by a n. If a1 1, a 3, a3 5,and an an 1 an an 3for all n 3, find the sum of the first thirty terms of the sequence. 300 b) 600 c) 900 d) Find the sum of the finite series b) 10 c) 16 d) The numbers 1,, 3,, 1000 are written in a row. Gonzo started at 1 and circled every 4 th number in red. Jake started at 1 and circled every 15 th number in blue. What is the smallest possible positive difference between a red number and a blue number? 4 b) 3 c) d) 1 Page of 6

3 Alpha Sequences and Series FAMAT State Convention If 100 more than the sum of n consecutive positive integers is equal to the sum of the next n consecutive positive integers, find n. 8 b) 10 c) 1 d) The sequence a i (for i=1,,3, ) is defined by setting a1 7 and, for i 1, taking ai to be the sum of the digits in the decimal representation of a. Find a b) 13 c) 16 d) 0 i Find the value of c for which the roots of 3 x x x c form an arithmetic progression. 16 b) 4 c) 36 d) The sequence t n of triangular numbers is defined for integers of the first 100 triangular numbers are multiples of 3? 1 n 1 by tn n( n 1). How many 33 b) 45 d) 60 d) Find the value of sin 10 sin 0 sin sin b) c) 4 d) 5 Page 3 of 6

4 Alpha Sequences and Series FAMAT State Convention Once upon a time, there were seven forests each housing seven dragons. Each dragon killed seven sheep. If left alive, each sheep would have eaten seven ears of corn. When not eaten, each ear of corn produced seven pounds of grain. How many pounds of grain were saved due to the existence of the forests? 401 b) c) d) The sum of the first 6 terms of an arithmetic progression whose first term is 1 is equal to the sum of the first 6 terms of the geometric progression beginning 1,, 4, Find the 6 th term of the arithmetic progression. 0 b) 4 c) 30 d) If the n k x n k 1 lim, then x is? 1 3 b) 1 c) 3 d) Find the sum of the infinite series b) 1 8 c) 7 3 d) In the harmonic sequence 6, 3,, 3, 6,..., what will the 8 th term be? b) 4 3 c) 8 9 d) 7 1 Page 4 of 6

5 Alpha Sequences and Series FAMAT State Convention If n is a multiple of 4, evaluate the sum (in terms of i) 1 i 3 i... ( n 1) i n, where i 1. ( ni) b) ( i ) c) ( ni ) d) ( n ni ) Simplify the product n 1 b) c) 3 n n n d) 4 n 3. The real roots of roots. 3 x 7x kx 0 are in geometric progression. Find the sum of these three.5 b) 3 c) 3.5 d) 4 4. Three numbers a, b, and c, none zero, form an arithmetic progression. Increasing a by 1 or increasing c by results in a geometric progression. Find b. 10 b) 1 c) 0 d) 4 5. The sum of the first 3 terms of a geometric sequence of positive integers is equal to 7 times the first term, and the sum of the first four terms is 45. What is the first term of this sequence? 3 b) 5 c) 7 d) 9 Page 5 of 6

6 Alpha Sequences and Series FAMAT State Convention The measures of the interior angles of a convex polygon of n sides are in arithmetic progression. If the common difference is5 and the largest angle is160, what is n? 7 b) 9 c) 10 d) The radius of the first circle is 1 inch, that of the nd is ½ inch, that of the 3 rd is ¼ inch and so on indefinitely. What is the sum of the areas of all the circles? b) 4 c) 7 d) The arithmetic mean of two positive numbers exceeds their geometric mean by 50. By how much does the square root of the larger of the two numbers exceed the square root of the smaller? 5 b) 10 c) 0 d) The sequence n1, n, n3,... consists of the number 1 and all those natural numbers greater than whose prime factors consist of only s, 3 s, and 5 s. Find the infinite sum:... n n n b) 15 4 c) 3 3 d) After all these questions, I think you need a break. The answer to this question is the same as the sum of the first 1000 positive even numbers b) c) d) Page 6 of 6

High School Mathematics Contest Spring 2006 Draft March 27, 2006

High School Mathematics Contest Spring 2006 Draft March 27, 2006 1. Going into the final exam, which will count as two tests, Courtney has test scores of 80, 81, 73, 65 and 91. What score does Courtney