An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 2 } + a _ { 4 } = 8$ and $a _ { 7 } = 52$. Find the common difference. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 6 and common difference $d$, let $S _ { n }$ denote the sum of the first $n$ terms. When
$$\frac { a _ { 8 } - a _ { 6 } } { S _ { 8 } - S _ { 6 } } = 2$$
holds, what is the value of $d$? [3 points]
(1) $-1$
(2) $-2$
(3) $-3$
(4) $-4$
(5) $-5$

For an arithmetic sequence $\left\{ a _ { n } \right\}$, when $a _ { 8 } - a _ { 4 } = 28$, find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, $$2 \left( a _ { 2 } + a _ { 3 } \right) = a _ { 9 }$$ Find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]

Let $S$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 4 } + a _ { 5 } = 24$ and $S _ { 6 } = 48$, then the common difference of $\left\{ a _ { n } \right\}$ is
A. 1
B. 2
C. 4
D. 8

Let $S _ { n }$ denote the sum of the first $n$ terms of the arithmetic sequence $\left\{ a _ { n } \right\}$. If $2 S _ { 1 } = 3 S _ { 2 } + 6$ , then the common difference $d = $ $\_\_\_\_$ .

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 7 }$

The common difference of the A.P. $b_{1},b_{2},\ldots,b_{m}$ is 2 more than common difference of A.P. $\mathrm{a}_{1},\mathrm{a}_{2},\ldots,\mathrm{a}_{\mathrm{n}}$. If $\mathrm{a}_{40}=-159,\mathrm{a}_{100}=-399$ and $\mathrm{b}_{100}=\mathrm{a}_{70}$, then $\mathrm{b}_{1}$ is equal to:
(1) 81
(2) $-127$
(3) $-81$
(4) 127

For an arithmetic sequence $(a_{n})$ with common difference $r$
$$\begin{aligned} & a_{30} - a_{25} < 53 \\ & a_{25} - a_{8} > 53 \end{aligned}$$
the inequalities are given.
What is the sum of the integer values that $r$ can take?
A) 49 B) 56 C) 64 D) 72 E) 84