Sequences and series › Arithmetic sequences

Arithmetic and geometric together

Positive x in a sequence of squares ●○○○○ 10 ptsThe numbers $1^2$ , $x^2$ , $3^2$ are consecutive terms of an arithmetic sequence. Find the positive value of $x$ . Round to two decimal places.
Middle term of an arithmetic sequence ●○○○○ 10 ptsIn the arithmetic sequence $3^2, x, 3^4$ , find the value of $x$ .
Harmonic numbers ●○○○○ 10 ptsCall three numbers $a, b, c$ harmonic when their reciprocals form an arithmetic sequence, that is, $\frac{1}{a}-\frac{1}{b}=\frac{1}{b}-\frac{1}{c}$ . For example, $15, 12, 10$ are harmonic because $\frac{1}{12}-\frac{1}{15}=\frac{1}{10}-\frac{1}{12}$ . Given that $x$ , $5$ , $3$ are harmonic numbers in this order with $x>5$ , find $x$ .
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 $B$ .
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?
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.
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 $a+b+c$ .
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.

More in Arithmetic sequences

First term and common difference Inserting means Ratios of two progressions Sum of an arithmetic sequence The nth term Which term? How many terms?