Practice (20)

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Let $a_1, a_2,\cdots, a_n > 0, n\ge 2,$ and $a_1+a_2+\cdots+a_n=1$. Prove $$\frac{a_1}{2-a_1} + \frac{a_2}{2-a_2}+\cdots+\frac{a_n}{2-a_n}\ge\frac{n}{2n-1}$$

Suppose all the terms in a geometric sequence $\{a_n\}$ are positive. If $|a_2-a_3|=14$ and $|a_1a_2a_3|=343$, find $a_5$.

Suppose no term in an arithmetic sequence $\{a_n\}$ equals $0$. Let $S_n$ be the sum of its first $n$ terms. If $S_{2n-1} = a_n^2$, find the expression for its $n^{th}$ term $a_n$.

Let $S_n$ be the sum of first $n$ terms in sequence $\{a_n\}$ where $$a_n=\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$$ Find $\lfloor{S_n}\rfloor$ where the floor function $\lfloor{x}\rfloor$ returns the largest integer not exceeding $x$.

Let sequence $\{a_n\}$ satisfy $a_0=1$ and $a_n=\frac{\sqrt{1+a_{n-1}^2}-1}{a_{n-1}}$. Prove $a_n > \frac{\pi}{2^{n+2}}$.

Show that $1+3+6+\cdots+\frac{n(n+1)}{2}=\frac{n(n+1)(n+2)}{6}$.

Show that $1+4+7+\cdots+(3n-2)=\frac{n(3n-1)}{2}$

Show that $1^2 + 3^2 + \cdots + (2n-1)^2=\frac{n(2n-1)(2n+1)}{3}$

Show that $1+5+9+\cdots+(4n-3)=2n^2 -n$

Show that $2+2^3 + 2^5+\cdots+2^{2n-1}=\frac{2(2^{2n}-1)}{3}$.

Show that $4^3 + 8^3 + 12^3 + \cdots + (4(k+1))^3=16(k+1)^2(k+2)^2$.

Show that $\frac{1}{5^2}+\frac{1}{5^4}+\cdots+\frac{1}{5^{2n}}=\frac{1}{24}(1-\frac{1}{25^n})$.

Show that $2^{-1}+2^{-2}+2^{-3}+\cdots+2^{-n}=1-2^{-n}$.

Show that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{3^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n}$

Determine all pairs $(a, b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression.

Let $$f(r) = \displaystyle\sum_{j=2}^{2008}\frac{1}{j^r} = \frac{1}{2^r}+\frac{1}{3^r}+\cdots+\frac{1}{2016^r}$$ Find $$\sum_{k=2}^{\infty}f(k)$$

Compute $$S_n=\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\cdots+\frac{n+1}{2^n}$$

Is it possible for a geometric sequence to contain three distinct prime numbers?

Is it possible to construct 12 geometric sequences to contain all the prime between 1 and 100?

Let $S_n$ be the sum of first $n$ terms of an arithmetic sequence. If $S_n=30$ and $S_{2n}=100$, compute $S_{3n}$.

Let $d\ne 0$ be the common difference of an arithmetic sequence $\{a_n\}$, and positive rational number $q < 1$ be the common ratio of a geometric sequence $\{b_n\}$. If $a_1=d$, $b_1=d^2$, and $\frac{a_1^2+a_2^2+a_3^2}{b_1+b_2+b_3}$ is a positive integer, what is the value of $q$?

Let $S_n$ be the sum of the first $n$ terms in geometric sequence $\{a_n\}$. If all $a_n$ are real numbers and $S_{10}=10$, and $S_{30}=70$, compute $S_{40}$.

Expanding $$\Big(\sqrt{x}+\frac{1}{2\sqrt[4]{x}}\Big)^n$$ and arranging all the terms in descending order of $x$'s power, if the coefficients of the first three terms form an arithmetic sequence, how many terms with integer power of $x$ are there?

Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018}$ • $a_{2020}$ • $a_{2022}$.

Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.