QuestionAugust 24, 2025

Translate between the recursive and explicit definitions for each sequence SEE EXAMPLE 2 22 a_(n)= ) 2,n=1 a_(n-1)+2,ngt 1

Translate between the recursive and explicit definitions for each sequence SEE EXAMPLE 2 22 a_(n)= ) 2,n=1 a_(n-1)+2,ngt 1
Translate between the recursive and explicit definitions for each sequence SEE EXAMPLE 2 22 a_(n)= ) 2,n=1 a_(n-1)+2,ngt 1

Solution
4.5(264 votes)

Answer

a_n = 2n Explanation 1. Identify the Recursive Pattern The recursive formula is a_n = a_{n-1} + 2 with a_1 = 2. This indicates an arithmetic sequence with a common difference of 2. 2. Derive the Explicit Formula For an arithmetic sequence, the explicit formula is given by **a_n = a_1 + (n-1) \cdot d**, where d is the common difference. Here, a_1 = 2 and d = 2. 3. Substitute Values into the Explicit Formula Substitute a_1 = 2 and d = 2 into the explicit formula: a_n = 2 + (n-1) \cdot 2. 4. Simplify the Expression Simplify to get a_n = 2 + 2n - 2 = 2n.

Explanation

1. Identify the Recursive Pattern<br /> The recursive formula is $a_n = a_{n-1} + 2$ with $a_1 = 2$. This indicates an arithmetic sequence with a common difference of 2.<br /><br />2. Derive the Explicit Formula<br /> For an arithmetic sequence, the explicit formula is given by **$a_n = a_1 + (n-1) \cdot d$**, where $d$ is the common difference. Here, $a_1 = 2$ and $d = 2$.<br /><br />3. Substitute Values into the Explicit Formula<br /> Substitute $a_1 = 2$ and $d = 2$ into the explicit formula: $a_n = 2 + (n-1) \cdot 2$.<br /><br />4. Simplify the Expression<br /> Simplify to get $a_n = 2 + 2n - 2 = 2n$.
Click to rate:

Similar Questions