Geometric Sequences: all problems

The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$. What is the eighth term of the sequence?

The third and fourth terms of an arithmetic sequence are the first and second terms of a geometric sequence. If the first two terms of the arithmetic sequence are $5, 2$, then what is the fourth term of the geometric sequence?

The numbers $a$, $b$, $c$, and $d$ form a geometric sequence, in that order. If $b$ is three more than $a$, and $c$ is nine more than $b$, what is the value of $d$?

The positive integers $A$, $B$, and $C$ form an arithmetic sequence while the integers $B$, $C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac53$, what is the smallest possible value of $A + B + C + D$?

Call a 3-digit number geometric if it has 3 distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

The sum of the first $2011$ terms of a geometric sequence is $200$. The sum of the first $4022$ terms is $380$. Find the sum of the first $6033$ terms.

Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S$?

Positive integers $a$, $b$, and $2009$, with $a < b < 2009$, form a geometric sequence with an integer ratio. What is $a$?

In 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.

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?

Let $p$ and $q$ be real numbers with $|p| < 1$ and $|q| < 1$ such that

Find $100pq$.

A sequence of positive integers with $a_1 = 1$ and $a_9 + a_{10} = 646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n \ge 1$, the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ are in geometric progression, and the terms $a_{2n}, a_{2n+1}$, and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than $1000$. Find $n + a_n$.

Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\frac{\sqrt{m} - n}{p}$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m + 10n + p$.