$a_n^{(1)}=-5 n \pm 2$
Answer (2)
The expression describes two sequences: a n ( 1 ) = − 5 n + 2 and a n ( 2 ) = − 5 n − 2 , both of which are arithmetic sequences with a common difference of − 5 . As n increases, both sequences diverge to negative infinity. Therefore, the final conclusion is that both sequences diverge towards − ∞ .
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The problem presents two sequences: a n = − 5 n + 2 and a n = − 5 n − 2 .
Both sequences are arithmetic with a common difference of − 5 .
As n approaches infinity, both sequences diverge to − ∞ .
Therefore, the sequences diverge to negative infinity: − ∞ .
Explanation
Understanding the Sequence We are given the sequence a n ( 1 ) = − 5 n ± 2 . This means we actually have two sequences to analyze: a n = − 5 n + 2 and a n = − 5 n − 2 . We will analyze them separately.
Analyzing the First Sequence Let's analyze the first sequence, a n = − 5 n + 2 . We can find the first few terms by substituting n = 1 , 2 , 3 , ... :
For n = 1 , a 1 = − 5 ( 1 ) + 2 = − 3 .
For n = 2 , a 2 = − 5 ( 2 ) + 2 = − 8 .
For n = 3 , a 3 = − 5 ( 3 ) + 2 = − 13 .
For n = 4 , a 4 = − 5 ( 4 ) + 2 = − 18 .
The difference between consecutive terms is constant: a 2 − a 1 = − 8 − ( − 3 ) = − 5 , a 3 − a 2 = − 13 − ( − 8 ) = − 5 , and so on. Therefore, this is an arithmetic sequence with a common difference of − 5 .
Analyzing the Second Sequence Now let's analyze the second sequence, a n = − 5 n − 2 . We can find the first few terms by substituting n = 1 , 2 , 3 , ... :
For n = 1 , a 1 = − 5 ( 1 ) − 2 = − 7 .
For n = 2 , a 2 = − 5 ( 2 ) − 2 = − 12 .
For n = 3 , a 3 = − 5 ( 3 ) − 2 = − 17 .
For n = 4 , a 4 = − 5 ( 4 ) − 2 = − 22 .
The difference between consecutive terms is constant: a 2 − a 1 = − 12 − ( − 7 ) = − 5 , a 3 − a 2 = − 17 − ( − 12 ) = − 5 , and so on. Therefore, this is also an arithmetic sequence with a common difference of − 5 .
Determining Convergence/Divergence Both sequences are arithmetic with a common difference of − 5 . As n increases, the terms of both sequences become more and more negative, tending towards negative infinity. Therefore, both sequences diverge to − ∞ .
Final Answer In summary, the given expression represents two arithmetic sequences: a n = − 5 n + 2 and a n = − 5 n − 2 . Both sequences diverge to negative infinity.
Examples
Arithmetic sequences are useful in calculating simple interest, where the interest earned each period is constant. For example, if you deposit money into an account that earns simple interest, the total amount in the account will increase by a fixed amount each period, forming an arithmetic sequence. Understanding arithmetic sequences helps in predicting the future value of such investments. Another example is in uniformly accelerated motion, where the velocity increases by a constant amount each second, forming an arithmetic sequence.
