Course Content
第四章:部分分式 Partial Fraction
0/1
第六章:角的形成及单位 Angles and Measurements
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第十三章: 方程组 Simultaneous Equations
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第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming
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高中一 | 高级数学
About Lesson

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.

*** QuickLaTeX cannot compile formula:
\[ \sum_{i=0}^{\infty}\frac{3i+2}{5^i}

*** Error message:
Missing $ inserted.
leading text: \end{document}

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}
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