# 进展问题（AP，GP，HP）详细介绍

2021年4月9日16:47:48 发表评论 1,029 次浏览

### 算术级数(AP)

AP的第n个项= a +(n-1)d

AP的" n"项总和= 0.5 n(第一项+最后一项)= 0.5 n [2a +(n-1)d]

### 几何级数(GP)

GP的n项= rn-1

GP的n个项之和(r <1)= [a(1-rn)] / [1-r]

GP的'n'个项的总和(r> 1)= [a(rn – 1)] / [r – 1]

GP的无限项总和(r <1)=(a)/(1 – r)

### 谐波渐进(HP)

• A≥G≥H
• A H = G2, 即A, G, H在GP中

### 样本问题

=>给定AP的第n个项= 11 +(n – 1)6

=>给定AP的第n个项= 5 + 6 n

=> n = 1->第一项= 5 + 6 = 11

=> n = 2->第二项= 5 + 12 = 17

=> n = 3->第三项= 5 + 18 = 23

=>给定AP的第n个项= 5 + 6 n

=>第一项= 5 + 6 = 11

=>第十项= 5 + 60 = 65

=> AP的10个项的总和= 0.5 n(第一项+最后一项)= 0.5 x 10(11 + 65)

=> AP的10个项的总和= 5 x 76 = 380

A =算术平均值=(4 + 6)/ 2 = 5

G =几何平均值=

= 4.8989

H =谐波均值=(2 x 4 x 6)/(4 + 6)= 48/10 = 4.8

=> GP的总和= 32 /(1 – 0.5)= 32 / 0.5 = 64

GP中三个数字的总和是26, 它们的乘积是216。

=>(a / r)+ a + a r = 26

=>一个(1 + r + r

2

)/ r = 26

=>(a / r)x(a)x(ar)= 216

=>一个

3

= 216

=>一个= 6

=> 6(1 + r + r

2

)/ r = 26

=>(1 + r + r

2

)/ r = 26/6 = 13/3

=> 3 + 3 r + 3 r

2

= 13 r

=> 3 r

2

– 10 r + 3 = 0

=>(r – 3)(r –(1/3))= 0

=> r = 3或r = 1/3

Nishant Arora