If $a$,$b$,$c \in {R^ + }$ are such that $2a$,$b$ and $4c$ are in $A$.$P$ and $c$,$a$ and $b$ are in $G$.$P$., then

Ten trucks, numbered $1$ to $10$ , are carrying packets of sugar. Each packet weights either $999\,g$ or $1000\,g$ and each truck carries only the packets equal weights. The combined weight of $1$ packet selected from the first truck,$2$ packets from the second,$4$ packets from the third, and so on, and $2^9$ packet from the tenth truck is $1022870\,g$. The trucks that have the lighter bags are

$0.14189189189….$ can be expressed as a rational number

Find the sum up to $20$ terms in the geometric progression $0.15,0.015,0.0015........$

Let $a_1, a_2, a_3, \ldots .$. be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, \ldots .$. be a sequence of positive integers in geometric progression with common ratio $2$ . If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality

$2\left(a_1+a_2+\ldots .+a_n\right)=b_1+b_2+\ldots . .+b_n$

holds for some positive integer $n$, is. . . . . . . 

If the first and the $n^{\text {th }}$ term of a $G.P.$ are $a$ and $b$, respectively, and if $P$ is the product of $n$ terms, prove that $P^{2}=(a b)^{n}$