The first term of an AP is a and the sum of the first $p$ terms is zero, show that the sum of its next $q$ terms is $\frac{-a(p+q) q}{p-1}$.
A man saved ₹ 66000 in 20 yr. In each succeeding year after the first year, he saved ₹ 200 more than what he saved in the previous year. How much did he save in the first year?
A man accepts a position with an initial salary of ₹ 5200 per month. It is understood that he will receive an automatic increase of ₹ 320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
If the $p$ th and $q$ th terms of a GP are $q$ and $p$ respectively, then show that its $(p+q)$ th term is $\left(\frac{q^p}{p^q}\right)^{\frac{1}{p-q}}$.
A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
The sum of interior angles of a triangle is $180^{\circ}$. Show that the sum of the interior angles of polygons with $3,4,5,6, \ldots$ sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
A side of an equilateral triangle is 20 cm long. A second equilateral triangle is inscribed in it by joining the mid-points of the sides of the first triangle. The process is continued as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.
In a potato race 20 potatoes are placed in a line at intervals of 4 m with the first potato 24 m from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
In a cricket tournament 16 school teams participated. A sum of ₹ 8000 is to be awarded among themselves as prize money. If the last placed team is awarded ₹ 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive?
If $a_1, a_2, a_3, \ldots, a_n$ are in AP, where $a_i>0$ for all $i$, show that $$ \frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}=\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}$$
Find the sum of the series
$$\left(3^3-2^3\right)+\left(5^3-4^3\right)+\left(7^3-6^3\right)+\supset \text { to (i) } n \text { terms. (ii) } 10 \text { terms. }$$
Find the rth term of an AP sum of whose first $n$ terms is $2 n+3 n^2$.
If $A$ is the arithmetic mean and $G_1, G_2$ be two geometric mean between any two numbers, then prove that $2 A=\frac{G_1^2}{G_2}+\frac{G_2^2}{G_1}$.
If $\theta_1, \theta_2, \theta_3, \ldots, \theta_n$ are in AP whose common difference is $d$, show that $$ \sec \theta_1 \sec \theta_2+\sec \theta_2 \sec \theta_3+\ldots+\sec \theta_{n-1} \sec \theta_n=\frac{\tan \theta_n-\tan \theta_1}{\sin d} $$
If the sum of $p$ terms of an AP is $q$ and the sum of $q$ terms is $p$, then show that the sum of $p+q$ terms is $-(p+q)$. Also, find the sum of first $p-q$ terms (where, $p>q$ ).
If $p$ th, $q$ th and $r$ th terms of an AP and GP are both and $c$ respectively, then show that $a^{b-c} \cdot b^{c-a} \cdot c^{a-b}=1$.
Match the following.
| Column I | Column II | ||
|---|---|---|---|
| (i) | $4,1, \frac{1}{4}, \frac{1}{16}$ | (a) | AP |
| (ii) | $2,3,5,7$ | (b) | Sequence |
| (iii) | $13,8,3,-2,-7$ | (c) | GP |
Match the following.
| Column I | Column II | ||
|---|---|---|---|
| (i) | $1^2+2^2+3^2+...+n^2$ | (a) | $\left[\frac{n(n+1)}{2}\right]^2$ |
| (ii) | $1^3+2^3+3^3+...+n^3$ | (b) | $n(n+1)$ |
| (iii) | $2+4+6+...+2n$ | (c) | $\frac{n(n+1)(2n+1)}{6}$ |
| (iv) | $1+2+3+...+n$ | (d) | $\frac{n(n+1)}{2}$ |
If the sum of $n$ terms of an AP is given by $S_n=3 n+2 n^2$, then the common difference of the AP is
If the third term of GP is 4 , then the product of its first 5 terms is
If 9 times the 9th term of an AP is equal to 13 times the 13th term, then the 22nd term of the AP is
If $x, 2 y$ and $3 z$ are in AP where the distinct numbers $x, y$ and $z$ are in GP, then the common ratio of the GP is
If in an $A P, S_n=q n^2$ and $S_m=q m^2$, where $S_r$ denotes the sum of $r$ terms of the $A P$, then $S_q$ equals to
Let $S_n$ denote the sum of the first $n$ terms of an AP, if $S_{2 n}=3 S_n$, then $S_{3 n}: S_n$ is equal to
The minimum value of $4^x+4^{1-x}, x \in R$ is
Let $S_n$ denote the sum of the cubes of the first $n$ natural numbers and $s_n$ denote the sum of the first $n$ natural numbers, then $\sum_\limits{r=1}^n \frac{S_r}{S_4}$ equals to
If $t_n$ denotes the $n$th term of the series $2+3+6+11+18+\ldots$, then $t_{50}$ is
The lengths of three unequal edges of a rectangular solid block are in GP. If the volume of the block is $216 \mathrm{~cm}^3$ and the total surface area is $252 \mathrm{~cm}^2$, then the length of the longest edge is
Two sequences cannot be in both AP and GP together.
Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.
Any term of an AP (except first) is equal to half the sum of terms which are equidistant from it.
The sum or difference of two GP, is again a GP.
If the sum of $n$ terms of a sequence is quadratic expression, then it always represents an AP.