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35 problems match “sequences”. Clear

Greatest third term of an arithmetic sequence ●●○○○ 20 ptsThe sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term?
Maximum leap years in 150 years ●●○○○ 20 ptsIn a calendar system in which leap years occur every four years, with no exceptions, what is the maximum possible number of leap years in a 150-year period?
Greatest number from two sequences ●●○○○ 20 ptsWhat is the greatest three-digit number $\overline{abc}$ such that $4, a, b$ form a geometric sequence and $b, c, 5$ form an arithmetic sequence? Here $a$ , $b$ , $c$ are digits.
Alternating sum in steps of four ●●○○○ 20 ptsFind the value of $2013-2009+2005-2001+\cdots+29-25$ .
Sixth term of a fraction sequence ●●○○○ 20 ptsFind the 6th term of the sequence $\frac{1}{3}, \frac{3}{5}, \frac{5}{8}, \frac{7}{12}, \frac{9}{17}, \ldots$ . Round your answer to two decimal places.
Erasing triples on a board ●●○○○ 20 ptsOn a board the numbers $1, 2, 3, \ldots, 27$ are written. One step consists of erasing three numbers $a, b, c$ from the board and writing in their place the number $a+b+c+n$ , where $n$ is a fixed positive integer. Determine the positive integer $n$ such that, after 13 steps, the number $n^2$ is written on the board.
Twentieth term of a radical sequence ●●○○○ 20 ptsConsider the sequence $\sqrt{2}$ , $-2$ , $\sqrt{6}$ , $-2\sqrt{2}$ , $\sqrt{10}$ , $-2\sqrt{3}$ , $\ldots$ . Following this pattern, find the 20th number in the sequence. Round to two decimal places.
Per-pass speed loss in aerobraking ●●○○○ 20 ptsA spacecraft approaches a planet at 18000 m/s and must slow to 8000 m/s over exactly 50 orbital passes. On each pass it keeps the same fraction of its speed. What percentage of its speed must it lose on each pass? Round to two decimal places.
Corrected value in a quadratic table ●●○○○ 20 ptsA student plots $y=ax^2+bx+c$ ( $a\neq 0$ ) using seven equally spaced $x$ -values $x_1<x_2<\cdots<x_7$ , where $x_{k+1}-x_k$ is constant. The recorded $y$ -values are $51, 107, 185, 285, 407, 549, 717$ for $x_1,\ldots,x_7$ respectively. Exactly one of these $y$ -values was calculated incorrectly. Find the correct value of that $y$ -value.
First term of a recurrence with a constant subtraction ●●○○○ 20 ptsA sequence $a_1, a_2, a_3, \ldots$ satisfies $a_n = 3a_{n-1} - x$ for every $n \ge 2$ , where $x$ is an integer that is the same for all terms. Given that $a_2 = 7$ and $a_4 = 43$ , find $a_1$ .