The sum of three consecutive terms in a geometric progression is $14$. If $1$ is added to the first and the second terms and $1$ is subtracted from the third, the resulting new terms are in arithmetic progression. Then the lowest of the original term is

Suppose $\log _a b+\log _b a=c$. The smallest possible integer value of $c$ for all $a, b>1$ is

If $a,\,b,\,c,\,d$ are positive real numbers such that $a + b + c + d$ $ = 2,$ then $M = (a + b)(c + d)$ satisfies the relation

When $\frac{1}{a} + \frac{1}{c} + \frac{1}{{a - b}} + \frac{1}{{c - d}} = 0$ and $b \ne a \ne c$, then $a,\;b,\;c$ are

Let $a, b, c, d\, \in \, R^+$ and $256\, abcd \geq  (a+b+c+d)^4$ and $3a + b + 2c + 5d = 11$ then $a^3 + b + c^2 + 5d$ is equal to :-

Suppose the sequence $a_1, a_2, a_3, \ldots$ is a n arithmetic progression of distinct numbers such that the sequence $a_1, a_2, a_4, a_8, \ldots$ is a geometric progression. The common ratio of the geometric progression is