91Ó°ÊÓ

An A.P. and a H.P., have the same first term, the same last term, and the same number of terms; prove that the product of the \(r\) th term from the beginning in one series and the \(r\) th term from the end in the other is independent of \(r\).

If the \((m+1)^{\text {th }},(n+1)^{\text {sh }}\) and \((r+1)^{t h}\) terms of an A.P. are in G.P., \(m, n, r\) are in H.P. show that the ratio of the common difference to the first term in the A.P. is \(-\frac{2}{n}\).

If \(S_{1}, S_{2}, S_{3}\) denote the sums of \(n\) terms of three A.P.'s whose first terms are unity and common differences in H.P., prove that \(n=\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}\).

If A.M between two numbers is 5 and their GM is 4, then find their HM.

Sum the series \(1+3+6+10+15+\ldots\) to \(n\) terms.

Sum to \(n\) terms the series and sum of infinite series \(\frac{1}{1 \cdot 3}+\frac{2}{1 \cdot 3 \cdot 5}+\frac{3}{1 \cdot 3 \cdot 5 \cdot 7}+\ldots\)