G Tewani Solutions for Chapter: Progression and Series, Exercise 1: DPP

If the first, fifth and last terms of an $\mathrm{A}.\mathrm{P}.$ are $l, m, p$ respectively, and sum of the $\mathrm{A}.\mathrm{P}.$ is $\frac{(l+p)(4p+m-5l)}{k(m-l)}$, then $k$ is

If $a_1, a_2, a_3,\dots ,a_{15}$ are in A.P. and $a_1+a_8+a_{15}=15$, then $a_2+a_3+a_8+a_{13}+a_{14}$ is equal to

If $a_1, a_2, a_3,\dots$ are in $\mathrm{A}.\mathrm{P}.$ and $a_i>0$ for each $i$, then $\sum _{i=1}^n\frac{n}{{\left(a_{i+1}\right)}^{\frac{2}{3}}+{\left(a_{i+1}\right)}^{\frac{1}{3}}{\left(a_i\right)}^{\frac{1}{3}}+{\left(a_i\right)}^{\frac{2}{3}}}$ is equal to

Between the numbers $2$ and $20, 8$ means are inserted, then their sum is

Let $a_1, a_2, a_3,\dots ,a_{4001}$ is an $A.P.$ such that $\frac{1}{a_1{a}_2}+\frac{1}{a_2{a}_3}+\dots +\frac{1}{a_{4000}{a}_{4001}}=10$ and $a_1+a_{4001}=50$, then $\left|a_1-a_{4001}\right|$ is equal to

An $\mathrm{A}.\mathrm{P}.$ consist of even number of terms $2n$ having middle terms equal to $1$ and $7$, respectively. If $n$ is the maximum value which satisfy $t_1{t}_{2n}+713\ge 0$, then the value of the first term of the series is

If the sum of the first $100$ terms of an $\mathrm{A}.\mathrm{P}.$ is $-1$ and the sum of even terms lying in first $100$ terms is $1,$ then which of the following is not true?

Given the sequence of numbers $x_1,x_2,x_3,x_4,\dots , x_{2005}$ which satisfy $\frac{x_1}{x_1+1}=\frac{x_2}{x_2+3}=\frac{x_3}{x_3+5}=\dots =\frac{x_{2005}}{x_{2005}+4009}$, the nature of the sequence is