Find the $9^{\text {th }}$ term in the following sequence whose $n^{\text {th }}$ term is $a_{n}=(-1)^{n-1} n^{3}$

If the sum of the series $54 + 51 + 48 + .............$ is $513$, then the number of terms are

Let the sequence ${a_1},{a_2},{a_3},.............{a_{2n}}$ form an $A.P. $ Then $a_1^2 - a_2^2 + a_3^3 - ......... + a_{2n - 1}^2 - a_{2n}^2 = $

The $8^{\text {th }}$ common term of the series $S _1=3+7+11+15+19+\ldots . .$ ; $S _2=1+6+11+16+21+\ldots .$ is $.......$.

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $\mathrm{S}_{10}=390$ and the ratio of the tenth and the fifth terms is $15: 7$, then $S_{15}-S_5$ is equal to:

If $a,\;b,\;c$ are in $A.P.$, then $\frac{{{{(a - c)}^2}}}{{({b^2} - ac)}} = $