The first term of an A.P. is \(a\), and the sum of the first \(p\) terms is zero. Show that the sum of its next \(q\) terms is
\[ -\dfrac{a(p+q)q}{p-1}. \]
\(-\dfrac{a(p+q)q}{p-1}\)
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
Rs 1400
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter.
(a) Find his salary for the tenth month.
(b) What is his total earnings during the first year?
Rs 8080 , Rs 83520
If the \(p\)th and \(q\)th terms of a G.P. are \(q\) and \(p\) respectively, show that its \((p+q)\)th term is
\[\left(\dfrac{q^p}{p^q}\right)^{\dfrac{1}{p-q}}.\]
\(\left(\dfrac{q^p}{p^q}\right)^{\tfrac{1}{p-q}}\)
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?
12 days
We know the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21-sided polygon.
3420°
A side of an equilateral triangle is 20 cm long. A second equilateral triangle is inscribed in it by joining the midpoints of the sides of the first triangle. The process is continued. Find the perimeter of the sixth inscribed equilateral triangle.
\(\dfrac{15}{8}\,\text{cm}\)
In a potato race 20 potatoes are placed in a line at intervals of 4 metres with the first potato 24 metres 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?
2480 m
In a cricket tournament 16 school teams participated. A sum of Rs 8000 is to be awarded among themselves as prize money. If the last placed team is awarded Rs 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?
Rs 725
If \(a_1, a_2, a_3, ..., a_n\) are in A.P., where \(a_i > 0\) for all \(i\), show that
\[ \dfrac{1}{\sqrt{a_1} + \sqrt{a_2}} + \dfrac{1}{\sqrt{a_2} + \sqrt{a_3}} + \cdots + \dfrac{1}{\sqrt{a_{n-1}} + \sqrt{a_n}} = \dfrac{n-1}{\sqrt{a_1} + \sqrt{a_n}}. \]
\(\dfrac{n-1}{\sqrt{a_1} + \sqrt{a_n}}\)
Find the sum of the series \((3^3 - 2^3) + (5^3 - 4^3) + (7^3 - 6^3) + ...\)
(i) to \(n\) terms
(ii) 10 terms
(i) \(4n^3 + 9n^2 + 6n\)
(ii) 4960
Find the \(r\)th term of an A.P. whose first \(n\) terms is \(2n + 3n^2\).
\(T_r = 6r - 1\)