Let a be the first term and r be the common ratio of the G.P.
∴ a = -3
It is known that, an = \(arn^-1\)
∴\(a^4\) = a\(r^3\)= (-3) \(r^3\)
\(a^2\) = a \(r^1\)= (-3) r
According to the given condition,
(-3) \(r^3\) = [(-3) r\(]^2\)
⇒ -3\(r^3\)= 9 \(r^2\)
⇒ r = -3
\(a^7\) = a\(r^7-1\)
= a \(r^6\)
= (-3) (-3\()^6\)
= - (3\()^7\)
= -2187
Thus, the seventh term of the G.P. is -2187.
Let \( 0 < z < y < x \) be three real numbers such that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression and \( x, \sqrt{2}y, z \) are in a geometric progression. If \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \), then \( 3(x + y + z)^2 \) is equal to ____________.
Find the mean and variance for the following frequency distribution.
Classes | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
Frequencies | 5 | 8 | 15 | 16 | 6 |