Sum to N Terms of an A.P.

In the arithmetic progression whose common difference is non-zero, the sum of first$3n$ terms is equal to the sum of next $n$ terms. Then the ratio of the sum of the first $2n$ terms to the next $2n$ terms is

If the sum of $p$ terms of an$\mathrm{A}.\mathrm{P}.$ is $q$ and the sum of$q$ terms is $p$, then the sum of $p+q$ terms will be

The first and last terms of an $\mathrm{A}.\mathrm{P}.$ are$1$ and $11$ . If the sum of its terms is $36$, then the number of terms will be

If $S_n$ denotes the sum of first $n$ terms of an $\mathrm{A}.\mathrm{P}.<a_n>$such that $\frac{S_m}{S_n}=\frac{m^2}{n^2},\text{ then }\frac{a_m}{a_n}=$

Sum of all two digit numbers which when divided by $4$ yield unity as remainder is

If the sums of $n$ terms of two arithmetic progressions are in the ratio$2n+5: 3n+4,$then write the ratio of their $m$th terms.

If $a_1 ,a_2, a_3, \dots , a_n$are in $\mathrm{A}.\mathrm{P}$. with common difference $d$, then the sum of the series $\sin d \left[\sec a_1\sec a_2+\sec a_2\sec a_3+\dots .+\sec a_{n-1}\sec a_n]\right.$, is

Let $S_n$ denote the sum of first$n$ terms of an $\mathrm{A}.\mathrm{P}.$ If $S_{2n}=3S_n$, then $S_{3n}:S_n$ is equal to

If in an $\mathrm{A}.\mathrm{P}. S_n=n^{2}q\text{ and }{S}_m=m^{2}q$ where $S_r$ denotes the sum of $r$ terms of the $\mathrm{A}.\mathrm{P}.,$ then $S_q$

If in an $\mathrm{A}.\mathrm{P}.,$ $S_n=n^{2}p\text{ and }{S}_m=m^{2}p$ where $S_r$ denotes the sum of $r$ terms of the$\mathrm{A}.\mathrm{P}.,$ then $S_p$ is equal to

If, $S_1$ is the sum of an arithmetic progression of $n$ odd number of terms and $S_2$ the sum of the terms of the series in odd places, then$\frac{S_1}{S_2}=$

If the first, second and last term of an $\mathrm{A}.\mathrm{P}.$ are $a, b \mathrm{and}$$2a$ respectively, then its sum is

If the sum of first $n$ even natural numbers is equal to $k$ times the sum of first $n$ odd natural numbers, then $k=$

The first and last term of an$\mathrm{A}. \mathrm{P}.$ are $a$ and $l$ respectively. If $S$is the sum of all the terms of the $\mathrm{A}.\mathrm{P}.$ and the common difference is given by $\frac{l^2-a^2}{k-(l+a)},$ then $k=$

Let $S_n$ denote the sum of $n$ terms of an$\mathrm{A}.\mathrm{P}.$ whose first term is $a$. If the common difference$d$$=S_n-k S_{n-1}+S_{n-2}$, then find the value of $k$

How many terms of the $\mathrm{A}.\mathrm{P}.$ $-6,-\frac{11}{2},-5, \dots$ are needed to given the sum $-25$?

Find the ${\mathrm{r}}^{\mathrm{th}}$ term of an $\mathrm{A}.\mathrm{P}.$ the sum of whose first $\mathrm{n}$ terms is $3{\mathrm{n}}^2+2\mathrm{n}$.

Find the sum of the series: $3+5+7+6+9+12+9+13+17+\dots$ to $3\mathrm{n}$ terms.

Find the sum of first $\mathrm{n}$ odd natural numbers.

Find the sum of first $\mathrm{n}$ natural numbers.