Assertion Reasoning Quiz

Assertion: For two positive numbers, their arithmetic mean equals their geometric mean only when the two numbers are equal.
Reason: For any two positive numbers, \(A\ge G\), where \(A\) and \(G\) denote their A.M. and G.M.

Assertion: A culture containing \(50\) bacteria initially and tripling every \(2\) hours will contain \(4050\) bacteria after \(8\) hours.
Reason: Eight hours contain four growth intervals, so the population is \(50\cdot 3^4\).

Assertion: If \(a_1=1\) and \(a_n=a_{n-1}+2\) for \(n\ge 2\), then the sum of the first five terms is \(25\).
Reason: The sequence is a G.P. with common ratio \(2\).

Assertion (A): The \(5^{th}\) term of the geometric progression 2, 4, 8, ... is 32.
Reason (R): The \(n^{th}\) term of a Geometric Progression is given by \(a_n = ar^{n-1}\).

Assertion (A): Let \(S_n\) denote the sum of first \(n\) terms of a G.P. The ratio of the sum of first \(n\) terms to the sum of terms from \((n+1)^{th}\) to \((2n)^{th}\) is \(\frac{1}{r^n}\).
Reason (R): The terms from \((n+1)^{th}\) to \((2n)^{th}\) form a G.P. with the first term being \(ar^n\) and the same common ratio \(r\).

Assertion: The sum of the first four terms of \(3,-6,12,-24,\ldots\) is \(-15\).
Reason: The fifth term of this G.P. is \(48\).