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35 problems match “sequences”. Clear

Third term of a power-plus-constant sequence ●●○○○ 20 ptsLet $\{a_n\}$ be the sequence given by $a_n = a^n + m$ for positive integers $n$ , where $a < 0$ and $m$ are constants. Given $a_1 = 2$ and $a_2 = 4$ , find $a_3$ .
Arithmetic then geometric progression integers ●●●○○ 50 ptsThree distinct integers $a$ , $b$ , $c$ satisfy the following conditions: - $abc = 17955$ ; - $a$ , $b$ , $c$ are three consecutive terms of an arithmetic sequence, in that order; - $3a+b$ , $3b+c$ , $3c+a$ are three consecutive terms of a geometric sequence, in that order. Find $a+b+c$ .
Sum of alternate terms of a sequence ●●●○○ 50 ptsA sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?
Integer trapped between two sequences ●●●○○ 50 ptsLet $\{a_n\}$ and $\{b_n\}$ be sequences with $a_1=0$ and $b_1=2013$ , such that for every positive integer $n$ the triples $(a_n, a_{n+1}, b_n)$ and $(a_{n+1}, b_{n+1}, b_n)$ are arithmetic sequences. Find the unique positive integer $c$ such that $a_n < c < b_n$ for all positive integers $n$ .
Sum of four progression terms ●●●○○ 50 ptsIn an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30$ . Find the sum of the four terms.
Argument of a sum of cis terms ●●●○○ 50 ptsThe number $\operatorname{cis} 75^\circ + \operatorname{cis} 83^\circ + \operatorname{cis} 91^\circ + \dots + \operatorname{cis} 147^\circ$ is expressed in the form $r \operatorname{cis} \theta$ , where $r > 0$ and $0^\circ \le \theta < 360^\circ$ . Find $\theta$ in degrees.
Sequence with a quadratic shift relation ●●●○○ 50 ptsLet the sequence $a_0, a_1, a_2, \ldots$ satisfy $a_0=a_1=11$ and $a_{m+n}=\frac{1}{2}\left(a_{2m}+a_{2n}\right)-(m-n)^2$ for all integers $m, n \geqslant 0$ . Find $a_{45}$ .
Counting nine-term sequences with ratio constraints ●●●○○ 50 ptsThe sequence $\{a_n\}$ has nine terms, $a_1=a_9=1$ , and for each $i\in\{1,2,\ldots,8\}$ we have $\frac{a_{i+1}}{a_i}\in\left\{2,1,-\frac{1}{2}\right\}$ . How many such sequences are there?
Position of a fraction in a sequence ●●●○○ 50 ptsThe positive rational numbers are arranged in the order $\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots$ . Find the position number of the number $\frac{1989}{1949}$ in this sequence.
Non-decreasing rolls of a die ●●●○○ 50 ptsA fair die is tossed 6 times. What is the probability that on each toss, the number shown is greater than or equal to the number on the previous toss? Give your answer as a decimal rounded to four decimal places.