Let \(a_1, a_2, a_3, \ldots, a_{10}\) be in G.P. with \(a_i > 0\) for \(i = 1, 2, \ldots, 10\) and \(S\) be the set of pairs \((r, k)\), \(r, k \in N\) (the set of natural numbers) for which \[\begin{vmatrix} \log_e a_1^r a_2^k & \log_e a_3^r a_4^k & \log_e a_5^r a_4^k \\ \log_e a_4^r a_5^k & \log_e a_6^r a_7^k & \log_e a_6^r a_8^k \\ \log_e a_7^r a_8^k & \log_e a_9^r a_9^k & \log_e a_9^r a_{10}^k \end{vmatrix} = 0\] Then the number of elements in \(S\) is:
Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An–1 and Hn–1 have arithmetic, geometric and harmonic means as An, Gn, Hn respectively. Which one of the following statements is correct?
If \(x = \displaystyle\sum_{n=0}^{\infty} a^n\), \(y = \displaystyle\sum_{n=0}^{\infty} b^n\), \(z = \displaystyle\sum_{n=0}^{\infty} c^n\) where a, b, c are in AP and \(|a| < 1\), \(|b| < 1\), \(|c| < 1\), then x, y, z are in
Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An–1 and Hn–1 have arithmetic, geometric and harmonic means as An, Gn, Hn respectively. Which one of the following statements is correct?
Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An–1 and Hn–1 have arithmetic, geometric and harmonic means as An, Gn, Hn respectively. Which one of the following statements is correct?