Write the next 4 terms of the sequence given that [tex]$a=3$[/tex], and [tex]$a_n=a_{n-1}-2$[/tex]
Answer (1)
The sequence starts with a = 3 .
Each subsequent term is obtained by subtracting 2 from the previous term: a n = a n − 1 − 2 .
Calculate the next four terms: a 2 = 1 , a 3 = − 1 , a 4 = − 3 , a 5 = − 5 .
The next 4 terms of the sequence are 1 , − 1 , − 3 , − 5 .
Explanation
Understanding the Sequence We are given a sequence where the first term is a = 3 , and each subsequent term is obtained by subtracting 2 from the previous term. We need to find the next 4 terms of this sequence.
Finding the Second Term To find the second term, we use the recursive formula a n = a n − 1 − 2 with n = 2 and a 1 = 3 . So, a 2 = a 1 − 2 = 3 − 2 = 1 .
Finding the Third Term To find the third term, we use the recursive formula a n = a n − 1 − 2 with n = 3 and a 2 = 1 . So, a 3 = a 2 − 2 = 1 − 2 = − 1 .
Finding the Fourth Term To find the fourth term, we use the recursive formula a n = a n − 1 − 2 with n = 4 and a 3 = − 1 . So, a 4 = a 3 − 2 = − 1 − 2 = − 3 .
Finding the Fifth Term To find the fifth term, we use the recursive formula a n = a n − 1 − 2 with n = 5 and a 4 = − 3 . So, a 5 = a 4 − 2 = − 3 − 2 = − 5 .
Listing the Next 4 Terms Therefore, the next 4 terms of the sequence are 1, -1, -3, and -5.
Examples
Sequences like this can model situations where a quantity decreases by a constant amount over time. For example, if you start with $3 and spend $2 each day, the sequence represents the amount of money you have at the end of each day. Understanding sequences helps in predicting future values in various real-life scenarios, from financial planning to scientific modeling.
