7.2 Arithmetic sequences

Find the general forms of the following arithmetic sequences

(a) [math]a_{12}=a_3=7[/math]

(b) [math]a_4=d=2[/math]

Consider a sequence [math]\{a_n\}=\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \sqrt{50}, \cdots[/math]. Which of the following values is between 1000 and 2000?

Find the series of the following arithmetic sequences

(a) [math]a_7=10, a_1=-2[/math]. [math]S_{15}[/math]

(b) [math]a_{10}=30, a_6=18[/math]. [math]S_{50}[/math]

Let [math]\{a_n\}[/math] be an arithmetic sequence. Suppose [math]a_{12}=1[/math]. Find [math]S_{23}[/math]. (Hint: mid-term).

The summation of five consecutive even numbers is 600. Find these numbers. Write your answer in ascending order.

Let [math]\{a_n\}[/math] be an airthmetic sequence with [math]a_1=34[/math] and common difference [math]-3[/math]. Find the [math]n[/math] to maximize [math]S_n[/math].

Let [math]\{a_n\}[/math] be an arithmetic sequence and [math]k[/math] be a nonzero constant. Which of the following sequences are also arithmetic?

(Calculator) Let [math]\{a_n\}[/math] be an arithmetic sequence with common difference [math]-3[/math]. If [math]a_1=399[/math] and [math]2^{a_1}\times2^{a_2}\times\cdots\times2^{a_n}<1[/math], what is the minimum of [math]n[/math]?

Let [math]\{a_n\}[/math] and [math]\{b_n\}[/math] be two arithmetic sequences.

(a) If [math]\dfrac{a_n}{b_n}=\dfrac{2n+3}{3n+4}[/math]. Find [math]\dfrac{a_{101}}{b_{101}}[/math].

(b) If [math]\dfrac{a_1+a_2+\cdots+a_n}{b_1+b_2+\cdots+b_n}=\dfrac{2n+3}{3n+4}[/math]. Find [math]\dfrac{a_{101}}{b_{101}}[/math].

Let [math]\{a_n\}[/math] be an airthmetic sequence. If [math]a_1+a_2+\cdots+a_n=\dfrac{-11n+31n^2}{2}[/math], what is the common difference?

If [math]a_n[/math] is a sequence that satisfies [math]a_{10}=10[/math] and [math]a_{n+1}-a_n=-2[/math] for all [math]n\geq1[/math], which of the following statments is/are true?

If [math]a_n[/math] is a positive sequence that satisfies [math]a_1=12[/math] and the remainder of [math]a_k[/math] divided by 7 is always 5, which of the following statments is/are true?

Evaluate [math]-2+1+3+5+7+9+11+13+\cdots+199[/math]?

Let [math]\{a_n\}[/math] be an arithmetic sequence with [math]a_1=34[/math] and [math]d=-3[/math]. Find the constant [math]n[/math] such that [math]S_n=-12[/math].

Given a sequence determined by [math]a_n=1-2n[/math], what is the summaion of the [math]501^{\text{st}}[/math] through the [math]1000^{\text{th}}[/math] terms of the sequence?

Insert 8 numbers between 1 and 12 to make the 10 numbers an arithmetic sequence. Find the common difference and the series.

(Calculator) Ashley donates money to Happy temple every year. On the first year, she donated 250 dollars. On the second year, she donated 255 dollars. On the third year, she donated 260 dollars, and so on. Obeying this pattern, how much will she have donated after 20 years?

On 11/1, Eliza and Charlie conduct a debate in a lecture hall in Happy City. There are 15 rows of seats in the lecture hall. Let [math]\{a_n\}[/math] be a sequence representing the number of seats in the [math]n^{\text{th}}[/math] row. Suppose [math]a_3=19, a_{10}=40[/math]. How many seats are there in the lecture hall?