Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } + a _ { 2 } = 10$ and $a _ { 4 } - a _ { 3 } = 2$.\n(I) Find the general term formula of $\left\{ a _ { n } \right\}$;\n(II) Let the geometric sequence $\left\{ b _ { n } \right\}$ satisfy $b _ { 2 } = a _ { 3 }$ and $b _ { 3 } = a _ { 7 }$. Question: Which term of the sequence $\left\{ a _ { n } \right\}$ is equal to $b _ { 6 }$?

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 3 } = 2$ and the sum of the first 3 terms $S _ { 3 } = \frac { 9 } { 2 }$ .
(I) Find the general term formula for $\left\{ a _ { n } \right\}$;
(II) A geometric sequence $\left\{ b _ { n } \right\}$ satisfies $b _ { 1 } = a _ { 1 } , b _ { 4 } = a _ { 15 }$. Find the sum of the first $n$ terms $T _ { n }$ of $\left\{ b _ { n } \right\}$.

17. (15 points) Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 2 , b _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ , $b _ { 1 } + \frac { 1 } { 2 } b _ { 2 } + \frac { 1 } { 3 } b _ { 3 } + \cdots + \frac { 1 } { n } b _ { n } = b _ { n + 1 } - 1 \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ .
(1) Find $a _ { n }$ and $b _ { n }$ ;
(2) Let $T _ { n }$ denote the sum of the first n terms of the sequence $\left\{ a _ { n } b _ { n } \right\}$ . Find $T _ { n }$ .

Let $S _ { n }$ be the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Find $S _ { n }$ and the minimum value of $S _ { n }$.

(12 points)
Let $S _ { n }$ be the sum of the first $n$ terms of arithmetic sequence $\{ a _ { n } \}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$.
(1) Find the general term formula of $\{ a _ { n } \}$;
(2) Find $S _ { n }$ and the minimum value of $S _ { n }$.

Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence with the sum of the first $n$ terms being $S _ { n }$. If $a _ { 2 } = - 3 , S _ { 5 } = - 10$, then $a _ { 5 } =$ $\_\_\_\_$, and the minimum value of $S _ { n }$ is $\_\_\_\_$.

17. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ with non-zero common difference. If $a _ { 3 } = S _ { 5 }$ and $a _ { 2 } a _ { 4 } = S _ { 4 }$ .
(1) Find the general term formula $a _ { n }$ of the sequence $\left\{ a _ { n } \right\}$ ;
(2) Find the minimum value of $n$ such that $S _ { n } > a _ { n }$ holds.
Answer: (1) $a _ { n } = 2 n - 6$ ; (2) 7 .

[Solution]

[Analysis] (1) From the given conditions, first find the value of $a _ { 3 }$ , then combine with the given conditions to find the common difference of the sequence to determine the general term formula;
(2) First find the expression for the sum of the first $n$ terms, then solve the quadratic inequality to determine the minimum value of $n$ . [Detailed Solution] (1) By the properties of arithmetic sequences, we have $S _ { 5 } = 5 a _ { 3 }$ , thus: $a _ { 3 } = 5 a _ { 3 }$ , therefore $a _ { 3 } = 0$ ,
Let the common difference of the arithmetic sequence be $d$ . Then: $a _ { 2 } a _ { 4 } = \left( a _ { 3 } - d \right) \left( a _ { 3 } + d \right) = - d ^ { 2 }$ , $S _ { 4 } = a _ { 1 } + a _ { 2 } + a _ { 3 } + a _ { 4 } = \left( a _ { 3 } - 2 d \right) + \left( a _ { 3 } - d \right) + a _ { 3 } + \left( a _ { 3 } + d \right) = 4a_3 - 2d = -2d$ ,
Thus: $- d ^ { 2 } = - 2 d$ . Since the common difference is non-zero, we have: $d = 2$ ,
The general term formula of the sequence is: $a _ { n } = a _ { 3 } + ( n - 3 ) d = 2 n - 6$ .
(2) From the general term formula, we have: $a _ { 1 } = 2 - 6 = - 4$ , thus: $S _ { n } = n \times ( - 4 ) + \frac { n ( n - 1 ) } { 2 } \times 2 = n ^ { 2 } - 5 n$ . The inequality $S _ { n } > a _ { n }$ becomes: $n ^ { 2 } - 5 n > 2 n - 6$ . Simplifying: $( n - 1 ) ( n - 6 ) > 0$ ,
Solving: $n < 1$ or $n > 6$ . Since $n$ is a positive integer, the minimum value of $n$ is 7 . [Key Point] The solution of basic quantities in arithmetic sequences is a fundamental problem in arithmetic sequences. The key to solving such problems is to master the relevant formulas of arithmetic sequences and apply them flexibly.

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$ .
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 } , a _ { 7 } , a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$ .

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$.
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 }$, $a _ { 7 }$, $a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$.