Arithmetic and Geometric

Tieria

2009-11-04 3:32 am

1. Find the number of terms and the first positive term in each of the following arithmetic sequences.
(a) -80, -68, -56, …, 124
(b) -64, -55, -46, …, 35

The answers are:
(a) number of terms = 18, first positive term = 4
(b) number of terms = 12, first positive term = 8

2. The sum of the 4th term and the 5th term of an arithmetic sequence is 18. If the sum of the first two terms is 42, find
(a) the general term,
(b) the 15th term.

The answers are (a) 27 – 4n, (b) -33.

※請列明計算步驟※

回答 (1)

用戶9112

2009-11-04 4:11 am

✔ 最佳答案

(a) The first term a = -80
The common difference d = -68 - (-80) = 12
The general term is -80 + (n - 1)(12) = 12n - 92
12n - 92 = 124
=> 12n = 216
=> n = 18
There are 18 terms
12n - 92 > 0
12n > 92
n > 7.67
The first positive term is the 8th term and the value is 12(8) - 92 = 4
(b) The first term a = -64
The common difference d = -55 - (-64) = 9
The general term is -64 + (n - 1)(9) = 9n - 73
9n - 73 = 35
=> 9n = 108
=> n = 12
There are 12 terms
9n - 73 > 0
9n > 73
n > 8.11
The first positive term is the 9th term and the value is 9(9) - 73 = 8
2. (a) Let the first term and the common difference be a and d respectively
a + (4 - 1)d + a + (5 - 1)d = 18
2a + 7d = 18 ... (1)
a + a + d = 42
2a + d = 42 ... (2)
(1) - (2) => 6d = - 24 => d = -4
Sub into (2), 2a - 4 = 42 => a = 23
The general term is 23 + (n - 1)(-4) = 27 - 4n
(b) the 15th term = 27 - 4(15) = -33

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