Arithmetic series and progressions

用戶8880

2007-02-12 10:55 am

1. find the sum of the integers from 501 to 1000 inclisive

2.find the sum of the arithmetic series
3+5+8+.....+398

3.two sequences U1,U2,U3,....are defined as follows
Sequence A: U1=2, Un+1=3-Un1
Sequence B: U1=2, Un+1=Un+1/2^n

更新1:

第3題唔記得打問 3a) for each of the two sequences, find the values of U2,U3,U4 and U5 3b)state for each sequence whether it is convergent, divergent or oscillating.

回答 (1)

myisland8132

2007-02-12 11:57 am

✔ 最佳答案

1. find the sum of the integers from 501 to 1000 inclusive
SUM
=n/2[2a+(n-1)d]
=500/2[2*501+(500-1)]
=250(1501)
=375250
2.find the sum of the arithmetic series
2+5+8+.....+398
let 2+3(n-1)=398
3(n-1)=396
n-1=132
n=133
SUM
=n/2[2a+(n-1)d]
=133/2[2*2+3(133-1)]
=26600
3.two sequences U1,U2,U3,....are defined as follows

Sequence A: U1=2, Un+1=3-Un1

Sequence B: U1=2, Un+1=Un+1/2^n

(a)

Sequence A

U1=2

U2=3-U1=3-2=1

U3=3-U2=3-1=2

U4=3-U3=3-2=1

U5=3-U4=3-1=2

Sequence B

U1=2

U2=U1+1/2=5/2

U3=U2+1/2^2=11/4

U4=U3+1/2^3=23/8

U5=U4+1/2^4=47/16

(b)

Sequence A is oscillating

Sequence B is convergence because

Un+1-Un=1/2^n

when n tends to infinity, the difference between two terms will tend to 0

So it is convergence.

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