If [tex]U_1=2[/tex] and [tex]U_n=U_{n-1}+n+3[/tex], for [tex]n \geq 2[/tex], find the values of [tex]U_2[/tex], [tex]U_3[/tex] and [tex]U_4[/tex].
Answer (1)
Calculate U 2 using the recursive formula: U 2 = U 1 + 2 + 3 = 2 + 5 = 7 .
Calculate U 3 using the recursive formula: U 3 = U 2 + 3 + 3 = 7 + 6 = 13 .
Calculate U 4 using the recursive formula: U 4 = U 3 + 4 + 3 = 13 + 7 = 20 .
The values are U 2 = 7 , U 3 = 13 , and U 4 = 20 , so the final answer is U 2 = 7 , U 3 = 13 , U 4 = 20 .
Explanation
Understanding the Problem We are given a recursive formula U n = U n − 1 + n + 3 for n g e 2 , and the initial value U 1 = 2 . Our goal is to find the values of U 2 , U 3 , and U 4 .
Calculating U2 To find U 2 , we substitute n = 2 into the recursive formula: U 2 = U 2 − 1 + 2 + 3 = U 1 + 5 Since U 1 = 2 , we have U 2 = 2 + 5 = 7
Calculating U3 To find U 3 , we substitute n = 3 into the recursive formula: U 3 = U 3 − 1 + 3 + 3 = U 2 + 6 Since U 2 = 7 , we have U 3 = 7 + 6 = 13
Calculating U4 To find U 4 , we substitute n = 4 into the recursive formula: U 4 = U 4 − 1 + 4 + 3 = U 3 + 7 Since U 3 = 13 , we have U 4 = 13 + 7 = 20
Final Answer Therefore, the values of U 2 , U 3 , and U 4 are 7, 13, and 20, respectively.
Examples
Recursive formulas are used in many real-world applications, such as calculating compound interest, modeling population growth, and designing computer algorithms. For example, if you invest $1000 in an account that earns 5% interest compounded annually, the amount of money in the account each year can be calculated using a recursive formula. Similarly, the Fibonacci sequence, where each term is the sum of the two preceding terms, is a classic example of a recursive sequence that appears in various natural phenomena, such as the branching of trees and the arrangement of leaves on a stem.
