Nishit K Sinha Solutions for Chapter: X+2 Maths, Exercise 3: Practice Exercises

The sum of the first $n$ terms of an AP is $\mathrm{n}(\mathrm{n}-1)$. Then, find the sum of the squares of these terms.

What is the sum of the following series? $7+26+63+124+\dots +999$

In $∆\mathrm{ABC}$, points ${\mathrm{P}}_1,{\mathrm{Q}}_1$, and ${\mathrm{R}}_1$ divide the lines $\mathrm{A} \mathrm{B}, \mathrm{B} \mathrm{C}$ and $\mathrm{A} \mathrm{C}$ respectively in the ratio of $2: 1$. In ${\mathrm{\Delta P}}_1{\mathrm{Q}}_1{\mathrm{R}}_1$, the points ${\mathrm{P}}_2,{\mathrm{Q}}_2$ and ${\mathrm{R}}_2$ divide the sides ${\mathrm{P}}_1{\mathrm{Q}}_1,{\mathrm{Q}}_1{\mathrm{R}}_1$ and ${\mathrm{P}}_1{\mathrm{R}}_1$ in the ratio of $2: 1$. In every such new triangle, a new triangle is generated by joining the points on the sides that divide these sides in the ratio of $2: 1$.Find the sum of the areas of all such triangles formed till infinity. (Area of $∆\mathrm{ABC}=$ ' $\mathrm{A}$ ' sq. units)

Find the sum of the series $-1+1^2-2+2^2-3+3^2$ $+\dots \mathrm{n}+{\mathrm{n}}^2$.

There are three numbers in an arithmetic progression. If the two larger numbers are increased by one, then the resulting numbers are prime. The product of these two primes and the smallest of the original numbers is $598$. Find the sum of the three numbers.

If three successive terms of a GP with the common ratio$\mathrm{r}>1$  form the sides of a triangle and $\left[\mathrm{r}\right]$ denotes the integral part of $\mathrm{x}$, then find $\left[\mathrm{r}\right]+[-\mathrm{r}]$.

In how many ways, can we select three natural numbers out of the first $10$natural numbers so that they are in a geometric progression with the common ratio greater than $1$?

Direction : Read the passage below and solve the questions based on it.

Let there be a series ' $\mathrm{S}$ ' with it is $\mathrm{nth}$ term be equal to $\mathrm{n}(\mathrm{x}{)}^{\mathrm{n}}$. Also, ${\mathrm{S}}_{\mathrm{n}}$ denotes the sum of the first $\mathrm{n}$ terms of the series $\mathrm{S}$.

What is ${\mathrm{S}}_5-{\mathrm{S}}_4$ equal to?