The sixth term of an $A.P.$ is equal to $2$, the value of the common difference of the $A.P.$ which makes the product ${a_1}{a_4}{a_5}$ least is given by
$x = \frac{8}{5}$
$x = \frac{5}{4}$
$x = 2/3$
None of these
Find the sum of all numbers between $200$ and $400$ which are divisible by $7.$
If $x_1 , x_2 , ..... , x_n$ and $\frac{1}{{{h_1}}},\frac{1}{{{h^2}}},......\frac{1}{{{h_n}}}$ are two $A.P' s$ such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$, then $x_5. h_{10}$ equals
A man starts repaying a loan as first instalment of $Rs.$ $100 .$ If he increases the instalment by $Rs \,5$ every month, what amount he will pay in the $30^{\text {th }}$ instalment?
Let the sequence $a_{n}$ be defined as follows:
${a_1} = 1,{a_n} = {a_{n - 1}} + 2$ for $n\, \ge \,2$
Find first five terms and write corresponding series.
Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f ( x )=\alpha x ^{5}+\beta x ^{3}+\gamma x , x \in R \quad$ and $\quad g : R \rightarrow R$ be such that $g(f(x))=x$ for all $x \in R$. If $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{n} \sum_{i=1}^{n} f\left(a_{i}\right)\right)\right)$ is equal to.