求救! 有關geometric sequence既題目

1) The first and the last term of a geometric sequence are 2 and 128 respectively. If the sum of all the terms of the sequence is 86, find the common ratio of the sequence.

Let r be the common ratio and let n be the number of terms.

The first term is a = 2.
The last term is arⁿ⁻¹ = 128 ⇒ rⁿ⁻¹ = 64
The sum of n terms is a(rⁿ - 1)/(r - 1) = 86
(rⁿ - 1)/(r - 1) = 43
rⁿ - 1 = 43(r - 1)
rⁿ⁻¹ - 1/r = 43(1 - 1/r) by dividing r on both sides
64 - 1/r = 43 - 43/r
42/r = -21
r = -2

The common ratio of the sequence is r = -2.


2) The common ratio and the sum to infinity of a geometric sequence are 2/3 and 2592 respectively.
(a) Find the first term of the sequence.

Let a be the first term.
It is given that the common ratio is r = 2/3 and the sum to infinity is 2592.
a/(1 - r) = 2592
a/(1 - 2/3) = 2592
a/(1/3) = 2592
3a = 2592
a = 864

(b) If the nth term of the sequence is 256,find
(i) the value of n,

The nth term of the sequence is arⁿ⁻¹ = 256
864(2/3)ⁿ⁻¹ = 256
(2/3)ⁿ⁻¹ = 8/27
n-1 = 3
n = 4

(II) the sum of the first n terms.

The sum of the first n terms is
　a(rⁿ - 1)/(r - 1)
= (864)[(2/3)⁴ - 1]/(2/3 - 1)
= (864)[1 - (2/3)⁴]/(1 - 2/3)
= (864)[1 - (2/3)⁴]/(1/3)
= 2592[1 - (2/3)⁴]
= 2080


3) The sum of the first 5 terms and the sum of the first 10 terms of a geometric sequence are 31 and 1023 respectively. Find the first term and the common ratio of the sequence.

Let the first term be a and the common ratio be r.
a(r⁵ - 1)/(r - 1) = 31 ...[1]

a(r¹⁰ - 1)/(r - 1) = 1023
a(r⁵ + 1)(r⁵ - 1)/(r - 1) = 1023 ...[2]

[2] ÷ [1] gives
r⁵ + 1 = 33
r⁵ = 32
r = 2

Put into [1],
a(2⁵ - 1)/(2 - 1) = 31
a(2⁵ - 1) = 31
a(32 - 1) = 31
a(31) = 31
a = 1

Therefore, the first term is a = 1 and the common ratio is r = 2.

2014-01-26 03:35:17 補充：
阿月～

謝謝你的意見～

☆ヾ(◕‿◕)ノ

2014-01-26 03:43:44 補充：
某程度上，我和你也複雜了。

其實用 r^n = 64r 就好了 ^___^

分享我Q1的做法：

Let r be the common ratio and n be the number of terms.

2r^(n-1)=128 -(1)

2(1-r^n) / 1-r =86
2-2r^n=86-86r
2r^n=86r-84 -(2)

(2)÷(1)：
r=(86r-84) / 128
128r=86r-84
84=-42r
r=-2

2014-01-26 02:03:50 補充：
第3題原來係 (r^10-1)=(r^5+1)(r^5-1)
諗唔到添...

2014-01-26 09:58:27 補充：
係啵~ Good~