91Ó°ÊÓ

Prove that if \(p, q, r(p \neq q)\) are terms (not necessarily consecutive) of an A.P., than there exists a rational number \(k\) such that \(\frac{(r-q)}{(q-p)}=k\).

Find the number of terms in the series \(20+19 \frac{1}{3}+18 \frac{2}{3}+\cdots \cdots\) of which the sum is 300 . Explain the double answer. Also find the maximum sum of the series.

Find the sum of all 2 digit odd numbers.

If \(\frac{3+5+7+\ldots \ldots+n \text { terms }}{5+8+11+\ldots \ldots+10 \text { terms }}=7\), then find the value of \(n\).

\(N\), the set of natural numbers, is partitioned into subsets \(S_{1}=\\{1\\}, S_{2}=\\{2,3\\}, S_{3}=\\{4,5,6\\}, S_{4}=\\{7,8,9\) \(10\\}\). Find the sum of the elements in the subset \(S_{50}\).

Find four numbers in A.P. whose sum is 32 and sum of squares is 276 .

150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day and so on. It takes 8 more days to finish the work now. Find the number of days in which the work was completed. \

Along a road lie an odd number of stones placed at intervals of 10 meters. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried the job with one of the end stones by carrying them in succession. In carrying all the stones he covered a distance of \(3 \mathrm{~km}\). Find the number of stones.

Balls are arranged in rows to from an equilateral triangle. The first row consists of one ball, the second row of two balls and so on. If 669 more balls are added then all the ball can be arranged in the shape of a square and each of the sides then contains 8 balls less then each side of the triangle did. Determine the initial numbers of balls.

Prove that the sum of the latter half of \(2 n\) terms of any A.P. is one-third the sum of \(3 n\) terms of the same A.P.