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8 problems match “geometric sequences”. Clear

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1. Fifth term of a recursive sequence ●○○○○ 10 ptsIn the sequence $\{a_n\}$, $a_1=2$ and $a_{n+1}=2a_n-1$. Find $a_5$.
2. Least difference between geometric and arithmetic terms ●●○○○ 20 ptsSequence $A$ is a geometric sequence and sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300$. Sequence $A$: $2, 4, 8, 16, 32, \ldots$. Sequence $B$: $20, 40, 60, 80, 100, \ldots$. Find the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B$.
3. 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.
4. 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.
5. 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 $a+b+c$.
6. 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 $n$.
7. 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.
8. Longest edge of a rectangular block ●●●○○ 50 ptsThe lengths of three unequal edges of a rectangular solid block are in geometric progression. The volume of the block is $216\ \text{cm}^3$ and its total surface area is $252\ \text{cm}^2$. Find the length of the longest edge, in cm.