EXERCISE - 1.7

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

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 $S?$

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 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 $S_4?$

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

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 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?

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 $31\left(r_1+r_2\right)?$