Special Sequences: all problems

What is the value of the product

Let $f(x) = x^2(1 - x)^2$. What is the value of the sum

Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

Rows $1$, $2$, $3$, $4$, and $5$ of a triangular array of integers are shown below:

Each row after the first row is formed by placing a $1$ at each end of the row, and each interior entry is $1$ greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the $2023$ numbers in the $2023$rd row?

Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?

If $T_n = 1 + 2 + 3 + \cdots + n$ and

for $n = 2, 3, 4, \ldots$, then $P_{1991}$ is closest to which of the following numbers?

Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum

of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Find the sum $\dfrac{1}{1(3)} + \dfrac{1}{3(5)} + \cdots + \dfrac{1}{(2n-1)(2n+1)} + \cdots + \dfrac{1}{255(257)}$.

Consider a decreasing sequence of $n$ positive integers $x_1 > x_2 > \cdots > x_n$ that satisfies the following conditions: The average of the first $3$ terms in the sequence is $2025$. For all $4 < k < n$, the average of the first $k$ terms is $1$ less than the average of the first $k - 1$ terms. What is the greatest possible value of $n$?

The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ starts with two $1$s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.

Evaluate

A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n-1} - a_{n-2}$ for each $n \ge 3$. What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985$, and the sum of the first $1985$ terms is $1492$?

Set $u_0 = \frac14$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence

This sequence tends to a limit; call it $L$. What is the least value of $k$ such that

The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$?

The sequence $\{a_n\}$ is defined by

The sequence $\{b_n\}$ is defined by

Find $\dfrac{b_{32}}{a_{32}}$.

The terms of the sequence $(a_i)$ defined by $a_{n+2} = \dfrac{a_n + 2009}{1 + a_{n+1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$.