Sequence and Series

There are $n$ balls in a box, and the balls are numbered $1, 2, 3,\dots , n$, respectively. One of the balls is removed from the box, and it turns out that the sum of the numbers on the remaining balls in the box is $5048$. If the number on the ball removed from the box is $m$, find the value of $m$

Let $a+a{r}_1+a{r}_1^2+a{r}_1^3+....$ and $a+a{r}_2+a{r}_2^2+a{r}_2^3+....$be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_1,$ and the sum of the second series is $r_2.$ What is $31\left(r_1+r_2\right)?$

The first four terms of an arithmetic sequence are $p, 9, \left(3p-q\right)$ and $\left(3p+q\right)$. What is the sum of digits of the ${2010}^{\mathrm{th}}$ term of the sequence?

Two non-decreasing sequences of non-negative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N.$ The smallest possible value of $N$ is $n.$ What is half of $n?$

The sequence ${\log }_{12}162, {\log }_{12}\mathrm{x}, {\log }_{12}y, {\log }_{12}z, {\log }_{12}1250$ is an arithmetic progression. What is $\frac{x}{10}?$

The sequence $S_1,{ S}_2,{ S}_3,\dots \dots S_{10}$ has the property that every term beginning with the third is the sum of the previous two. That is, $S_n=S_{n-2}+S_{n-1}$ for $n\ge 3$. Suppose that $S_9=110$ and $S_7=42$. What is $S_4?$

Let $a<b<c$ be three integers such that $a, b, c$ is an arithmetic progression and $a, c, b$ is a geometric progression. What is the smallest possible value of $10c?$

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 $S?$

If the minimum value of $\sqrt{\sum _{k=1}^{100}\left|n-k\right|}$, where $n$ ranges over all positive integers, is $m,$ find $\frac{m}{50}.$

A sequence $\left\{a_n\right\}$ is defined by $a_1=2, a_n=\frac{1+a_{n-1}}{1-a_{n-1}}, n\ge 2$. Find the value of $\left(1100+2008a_{2007}\right)$

Suppose $a_n$ denotes the last two digits of $7^n$. For example, $a_2=49, a_3=43$. The value of $a_1+a_2+a_3+\dots \dots ..+a_{2007}$ is given by $x$. Find sum of all the digits of $x$

What is the value of

$2100-\left(x+1\right)\left(x+2006\right)\left[\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\dots +\frac{1}{\left(x+2005\right)\left(x+2006\right)}\right]$?

James calculates the sum of the first $n$ positive integers and finds that the sum is $5053.$ If he has counted one integer twice, which one is it?

Consider a sequence of real numbers $\left\{a_n\right\}$ defined by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+n{a}_n}$ for $n\ge 1$.

Find the value of $\frac{1}{a_{2005}}-2009000$

Find the value of the positive integer $n$ if $\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\dots +\frac{1}{\sqrt{n}+\sqrt{n+1}}=5$

Let $x$ and $y$ be positive real numbers. What is the smallest possible value of $\frac{16}{x}+\frac{108}{y}+xy$?

Consider the sequence of numbers: $4, 7, 1, 8, 9, 7, 6,$... For $n>2,$ the $n^{\mathrm{th}}$ term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. If smallest value of $n$ for $Sn>10,000$ is $abcd$, then find $a+b+c+d$

In the sequence $2001, 2002, 2003,$..... each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001+2002-2003=2000.$ What is the ${2004}^{\mathrm{th}}$ term in this sequence?

A sequence of three real numbers forms an arithmetic progression with a first term of $9.$ If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the largest possible value for the third term in the geometric progression?

The increasing sequence of positive integers $a_1, a_2, a_3,$..... has the property that $a_{n+2}=a_n+a_{n+1}$ for all $n\ge 1.$ Suppose that $a_7=120.$ What is $a_8$?