Math topic中文

Arithmetic sequences and Gauss summation

Recognise arithmetic sequences; find the nth term and sum a sequence by pairing first and last terms (Gauss method).

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Expressing patterns with numbers or symbols

Practice questions

1已知数列 $\{a_{n}\}$ 的前 $n$ 项和为 $S_{n}$ ，首项为 $a_{1}$ ，且 $4$ ， $a_{n}$ ， $S_{n}$ 成等差数列． (1)求数列 $\{a_{n}\}$ 的通项公式； (2)若 $a_{n}^{2}={{2}^{{{b}_{n}}}}$ …
2已知数列 $\{a_{n}\}$ 为等差数列，且 $a_{1}+a_{7}+a_{13}=π$ ，则 $\tan (a_{2}+a_{12})$ 的值为 $()$ A. $\sqrt {3}$ B. $- \sqrt {3}$ …
3When Yunji added all the integers from $1$ through $9$ , she mistakenly left out a number. Her incorrect sum turned out t…
4Lucius is counting backward by 7s. His first three numbers are 100, 93, and 86. What is his 10th number? A. 30 B. 37 C.…
5已知等差数列 $\{a_n\}$ 中， $a_1 = 3$ ，公差 $d = 2$ ，求前10项和 $S_{10}$ 。
6The sum of the first n terms of a linear sequence is . Show that the sum of the first n terms in the sequence is always…
7The sum of 25 consecutive even integers is 10000. What is the largest of these 25 consecutive integers? A. 360 B. 388 C.…
8On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widge…
9Jami picked three equally spaced integer numbers on the number line. The sum of the first and the second numbers is 40,…
10已知数列 $\{a_{n}\}$ 满足 $a_{1}=1$ ，且 $a_{n+1}=\dfrac{2{{a}_{n}}}{3{{a}_{n}}+2}(n\in N^{*}).(1)$ 证明：数列 $\left\{\dfrac{2}{{{a}_{n}}}\right\}$ …
11In how many ways can 90 be written as the sum of two or more consecutive even positive integers that are arranged in inc…
12已知 $S_{n}$ 是公差不为零的等差数列 $\{a_{n}\}$ 的前 $n$ 项和， $4a_{1}a_{11}=(a_{2}+a_{4})^{2}$ ，且 $a_{4}+a_{5}=25$ ，则 $S_{20}=$ ___.
13已知在等差数列 $\{a{\,\!}_{n}\}$ 中， $S{\,\!}_{n}=33$ ， $S{\,\!}_{2n}=44$ ，则这个数列的前 $3n$ 项和 $S{\,\!}_{3n}$ 为___．
14已知数列 $\{a_{n}\}$ 中， $a_{n+1}-a_{n}=2$ ，若 $a_{1}=3$ ，则 $a_{10}=$ ___ .
15已知数列 $\{a_{n}\}$ 满足 $a_{n}+a_{n+1}=2n$ ， $S_{n}$ 为其前 $n$ 项和，若 $a_{1}=a_{4}$ ，则 $S_{101}=(\quad)$ A. $4882$ B. $5100$ …
16(1)三角形 $ABC$ 中， $AB=2\sqrt{3},BC=2,\angle C={{60}^{0}}$ ，则三角形 $ABC$ 的面积为___ (2)设等比数列 $\left\{ {{a}_{n}} \right\}$ 满足 $a{\,\!}_{1}+a{\,\!}_{3}=10$ …
17已知数列 $\{a_{n}\}$ 中， $a_{1}=5$ ，其前 $n$ 项和 $S_{n}$ 满足 ${S}_{6}={S}_{13}.$ 若 $S_{n}>0$ ，则 $n$ 的最大值为___.
18An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous…
19Fifteen integers $a_1, a_2, a_3, \cdots, a_{15}$ are arranged in order on a number line. The integers are equally spaced…