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 (2)
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 . We will do this by repeatedly applying the recursive formula.
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 analyzing financial investments. For example, if you invest a certain amount of money and earn interest each year, the amount of money you have each year can be modeled using a recursive formula. Understanding recursive formulas helps in predicting future outcomes based on current conditions.
The values calculated are U 2 = 7 , U 3 = 13 , and U 4 = 20 using the recursive formula provided. Each value was derived step-by-step by substituting into the formula. This method demonstrates how recursive sequences work by building each term from the previous one.
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