Question

Let a_n be the nth term of an arithmetic sequence, where the common difference is an integer. It is given that a_1 = 74, a_3 > 60 and a_8 − a_5 < −15.

Find the common difference of the sequence

David claims that there are 14 positive terms in the sequence. Do you agree? Explain your answer.

Collected in the board: Arithmetic sequence

Steven Zheng posted 1 year ago

Answer

The n-th term of an arithmetic sequence is represented as

a_n = a_1 + (n – 1)d.

Given a_1 = 74

Then

a_3 = 74+2d>60
2d>-14
d>-7
(1)
a_5 = 74+4d
a_8 = 74+7d
a_8-a_5 = 3d<-15
d<-5
(2)

Based on (1) and (2), the common difference is in the range of

-7< d< -5

Since the common difference is an integer, we get

d = -6

The general term formula for the sequence is

a_n = 74 - (n – 1)6 = 80-6n

If a_n > 0

80-6n > 0

n < \dfrac{80}{6} = \dfrac{40}{3} <14

Therefore, there are less 14 positive terms in the sequence

Steven Zheng posted 1 week ago

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