Assertion Reasoning Quiz

Assertion: If the A.M. and G.M. of two positive numbers are \(13\) and \(12\), respectively, then the numbers are \(10\) and \(16\).
Reason: For the two numbers \(x\) and \(y\), the given information implies \(x+y=26\) and \(xy=144\).

Assertion (A): If the \(p^{th}\), \(q^{th}\), and \(r^{th}\) terms of a G.P. are \(a\), \(b\), and \(c\) respectively, then \(a^{q-r} b^{r-p} c^{p-q} = 1\).
Reason (R): This relationship holds true only if \(p\), \(q\), and \(r\) are strictly in an Arithmetic Progression.

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): The sum of the sequence 7, 77, 777... to \(n\) terms is evaluated by directly applying the sum formula for a Geometric Progression.
Reason (R): The sequence 7, 77, 777... is not a G.P. itself, but can be rewritten as \(\frac{7}{9} [(10-1) + (10^2-1) + \dots]\) to form geometric series.

Assertion (A): If the A.M. and G.M. of two positive numbers are 10 and 8 respectively, the numbers are 16 and 4.
Reason (R): The two numbers are the roots of the quadratic equation \(x^2 - 2(A.M.)x + (G.M.)^2 = 0\).

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 \(r\).