F.5 Arithmetic and Geometric Sequences

紫魂

2008-09-25 9:42 am

1. The general term of an arithmetic sequence is given by T(n)= 8n+3. Find the sum of the following series.
T(1)+T(4)+T(7)+T(10)+....+T(64)

2. The common ratio of a geometric sequence is -2, the nth term is 8/3, the sum of the first n terms is 57/32. Find the value of n.

3. T(n) is the general term of the geometric sequence 1, 開方2, 2, 開方8.....
(a) Find the common ratio and the number of terms of the geometric sequence T(2), T(4), T(6),....,T(20)
(b) Find the sum of T(2)+T(4)+T(6)+......+T(20)

回答 (1)

Uncle Michael

2008-09-25 10:58 am

✔ 最佳答案

1.
Sum = T(1) + T(4) + T(7) + T(10) + ...... T(64)
Sum = 11 + 35 + 59 + 83 + ...... + 515

It is equal to the sum of an arithmetic sequence with
a = 11
d = 35 - 11 = 24
l = 515
n = (515-11)/24 + 1 = 22

Sum = n(a+l)/2
Sum = 22(11+515)/2
Sum = 5786

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2.
First term = a
Common ratio, r = -2

the nth term:
a(-2)n-1 = 8/3 ...... (1)

The sum of the first n terms:
a[1-(-2)n]/[1-(-2)] = 57/32
a[1-(-2)n]/3 = 57/32
a[1-(-2)n] = 171/32
a - a(-2)n = 171/32
a - a(-2)n-1(-2) = 171/32 ...... (2)

Put (1) into (2):
a - (8/3)(-2) = 171/32
a = 171/32 - 16/3
a = 1/96

Put a = 1/96 into (1)
(1/96)(-2)n-1 = 8/3
(-2)n-1 = 256
(-2)n-1 = (-2)8
n - 1 = 8
n = 9

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3.(a)
1, √2, 2, √8 ......

Common ratio = √2

Number of terms = (20 - 2)/2 + 1
Number of terms = 10

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3.(b)
Sum = T(2) + T(4) + T(6) + ...... + T(20)
Sum = √2 + √8 + √32 + ...... + T(20)

It is equal to the sum of a geometric sequence with
a = √2
r = 2
n = 10

Sum = (√2)(210-1)/(2-1)
Sum = 1023√2
=

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