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第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

16.5 简易特殊数列的和 The Sum of Some Simple Progression

16.5 The Sum of Some Simple Progression 简易特殊数列的和


Calculate the sum of the following series

计算下列各级数的和

1.

2 x (-2) + 5 x 2 + 10 x 6 + 17 x 10 + . . . + (n^2 + 1)(4n – 6)

Answer:
n^4 – 5n
2.

1 x 1 + (-1) x 5 + (-3) x 9 + (-5) x 13 + …. + (-37) x 77

Answer:
-19360
3.

2^2 x 2 + 3^2 x 5 + 4^2 x 8 + 5^2 x 11 + … + 11^2 x 29

Answer:
11045
4.

1 x 4 x 7 + 2 x 5 x 8 + 3 x 6 x 9 + … + 10 x 13 x 16

Answer:
7480
5.

18^2 – 17^2 + 16^2 – 15^2 + 14^2 – 13^2 + … + 2^2 – 1^2

Answer:
171
6.

1 + \frac{4}{5} + \frac{7}{25} + \frac{10}{125} + …

Answer:
\frac{35}{16}
7.

1 + \frac{3}{3} + \frac{7}{9} + \frac{15}{27} + \cdots

Answer:
4\frac{1}{2}
8.

1 + \frac{5}{2} + \frac{9}{4} + \frac{13}{8} + \cdots to n term(至第n项)

Answer:
10 – \frac{1}{2^{n-4}}\frac{4n-3}{2^{n-1}}

9.

1 + (1 + a)r + (1 + a + a^2)r^2 + (1 + a + a^2 + a^3)r^3 + … where(其中) |ar| < 1.

Answer:
\frac{1}{(1-ar)(1-r)}
10.

\[ \sum_{i=0}^{\infty}\frac{3i+2}{5^i}

Answer:
\frac{55}{16}
11.

\frac{1}{2\times3} + \frac{1}{3\times4} + \frac{1}{4\times5} + \frac{1}{5\times6} + \frac{1}{6\times7} + \cdot\cdot\cdot + \frac{1}{51\times52}

Answer:
\frac{25}{52}
12.

\frac{1}{2\cdot4} + \frac{1}{4\cdot6} + \frac{1}{6\cdot8} + \cdot\cdot\cdot

Answer:
\frac{1}{4}
13.

\sum \limits_{i=1}^{n}\frac{1}{(2i-1)(2i+1)}

Answer:
\frac{n}{2n+1}
14.

\frac{1}{1\times4} + \frac{1}{2\times5} + \frac{1}{3\times6} + \cdot\cdot\cdot + \frac{1}{20\times23}

Answer:
\frac{119}{253}
15.

Simplify \sum (3n – 2)^2. Hence, find the value of 732 + 762 + 792 + … + 2082 .
化简\sum (3n – 2)^2. 据此, 求 732 + 762 + 792 + … + 2082 的值。

Answer:
3n^3\frac{3}{2}n^2\frac{1}{2}n , 981019
16.

Define a series such that, the first term is 3, the second term is 3 + 8, the third term is 3 + 8 + 13, the fourth term is 3 + 8 + 13 + 18 and so on. Find the sum of the first n terms of this series.

一级数其首项是3, 第二项是 3 + 8, 第三项是 3 + 8 + 13, 第四项是 3 + 8 + 13 + 18 以此类推。求这个级数首n 项的和。

Answer:
\frac{n(n+1)(10n+8)}{12}