Embibe Experts Solutions for Chapter: Algebra, Exercise 9: EXERCISE - 1.9

The sequence $a_1, a_2, a_3, \dots ,$ satisfies $a_1=19$ and $a_9=99.$ Also, $a_n$ is the arithmetic mean of the first $n-1$ terms whenever $n\ge 3.$ What is $a_2?$

For positive integers $m$ and $n$ such that $m+10<n+1,$ both the mean and the median of the set $\{m,m+4,m+10,n+1,n+2,2n\}$ are equal to $n.$ What is $m+n?$

A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. The least number of distinct values that can occur in the list is a three digit number $xyz.$ What is ${\left(y+z\right)}^2?$

$A$, $B, C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40 \mathrm{kg},$ the mean weight of the rocks in $B$ is $50 \mathrm{kg},$ the mean weight of the rocks in the combined piles $A$ and $B$ is $43 \mathrm{kg},$ and the mean weight of the rocks in the combined piles $A$ and $C$ is $44 \mathrm{kg}.$ What is the greatest possible integer value for the mean in $\mathrm{kg}$ of the rocks in the combined piles $B$ and $C?$

On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?

The average (arithmetic mean) age of a group consisting of doctors and lawyers is $40.$ If the doctors average $35$ and the lawyers $50$ years old, then the ratio of the number of doctors to the number of lawyers is $\mathrm{K}:1.$ Find $K.$

Last year Sunita took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100,$ inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95.$ If her score on the sixth test was $S.$ Find $\frac{S}{4}.$

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x?$