Arun Sharma Solutions for Exercise 2: Level of Difficulty

If a man saves $₹ 1000$ each year and invests at the end of the year at $5\%$ compound interest, how much will the amount be at the end of $15$ years?

Find the infinite sum of the series $\frac{1}{1}+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+...$.

The sum of the series $3\times 5+5\times 7+7\times 9+...$ up to $\mathrm{n}$ terms is _____.

The sum of the series $\frac{1}{3}+\frac{4}{15}+\frac{4}{35}+\frac{4}{63}+...$ up to $6$ terms is _____.

The sum of the series $\frac{1}{3}+\frac{4}{15}+\frac{4}{35}+\frac{4}{63}+...$ up to infinite terms is _____.

Rahul drew a rectangular grid of $625$ cells, arranged in $25$ rows and $25$ columns, and filled each cell with a number. The numbers with which he filled each cell were such that the numbers of each row taken from left to right formed an arithmetic series, and the numbers of each column taken from top to bottom also formed an arithmetic series. The $6$th and the $20$th numbers of the $5$th row are $37$ and $73$, respectively, while the $6$th and the $20$th numbers of the $25$th row are $63$ and $87$, respectively. What is the sum of all the numbers in the grid?

How many four-digit numbers have the property that their digits, when taken from left to right, form an arithmetic or a geometric progression? Assume that all the digits are distinct.

An arithmetic progression $\mathrm{P}$ consists of $\mathrm{n}$ terms. From this progression, three different progressions ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, and ${\mathrm{P}}_3$ are created such that ${\mathrm{P}}_1$ is obtained by the $1$st, $4$th, $7$th, $...$, terms of $\mathrm{P}$, ${\mathrm{P}}_2$ is obtained by the $2$nd, $5$th, $8$th, $...$, terms of $\mathrm{P}$, while ${\mathrm{P}}_3$ is obtained by the $3$rd, $6$th, $9$th, $...$, terms of $\mathrm{P}$. It is found that amongst ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, and ${\mathrm{P}}_3$, two progressions have the property that their average itself is a term of the original progression $\mathrm{P}$. Which of the following can be a possible value of $\mathrm{n}$?