#### Practice (20)

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If the sum of an arithmetic progression of six positive integer terms is 78, what is the greatest possible difference between consecutive terms?

When $\frac{1}{98}$ is expressed as a decimal, what is the $10^{th}$ digit to the right of the decimal point?

The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?

The sum of six consecutive positive integers is 2013. What is the largest of these six integers?

How many terms are in the arithmetic sequence $13$, $16$, $19$, $\dotsc$, $70$, $73$?

Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,\!000$th number in the sequence?

When counting from $3$ to $201$, $53$ is the $51^{st}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{th}$ number counted. What is $n$?

Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying $1"$, so Blair follows by saying $"1, 2"$. Jo then says $"1, 2, 3"$, and so on. What is the $53^{\text{rd}}$ number said?

The real numbers $c,b,a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$ The quadratic $ax^2+bx+c$ has exactly one root. What is this root?

Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?

In the eight-term sequence $A,B,C,D,E,F,G,H$, the value of $C$ is 5 and the sum of any three consecutive terms is 30. What is $A+H$?

Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?

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?

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

On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

The arithmetic mean of 11 numbers is 78. If 1 is subtracted from the first, 2 is subtracted from the second, 3 is subtracted from the third, and so forth, until 11 is subtracted from the eleventh, what is the arithmetic mean of the 11 resulting numbers?

Alex added the page numbers of a book together and got a total of 888. Unfortunately, he didn't notice that one of the sheets of the book was missing with an odd page number on the front and an even page number on the back. What was the page number on the final page in the book?

The figure shows the first three stages of a fractal, respectively. We must find how many circles in Stage 5 of the fractal.

The sum of the first $n$ terms of a sequence, $a_1 + a_2 + \cdots + a_n$, is given by the formula $S_n = n^2 + 4n + 8$ What is the value of $a_6$?

Seven consecutive positive integers have a sum of 91. So what is the largest of these integers?

We are asked to find the largest sum of calendar dates for seven consecutive Fridays in any given year.

Evaluate $\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}$.