Question

what is the general term equation, a(n), for the arithmetic sequence 21, 25, 29, 33... and what is the 21st term of this sequence? hint: ( a(n) = a(1) + d(n - 1) ), where ( a(1) ) is the first term and ( d ) is the common difference. ( \bigcirc a(n) = 9 - 4n; a(21) = -75 ) ( \bigcirc a(n) = 17 - 4n; a(21) = -67 ) ( \bigcirc a(n) = 13 - 4n; a(21) = -71 ) ( \bigcirc a(n) = 17 + 4n; a(21) = 101 )

Explanation:

Step1: Identify first term and common difference

The first term \( A(1) \) of the arithmetic sequence \( 21, 25, 29, 33, \dots \) is \( 21 \). The common difference \( d \) is \( 25 - 21 = 4 \).

Step2: Use the arithmetic sequence formula

The formula for the \( n \)-th term of an arithmetic sequence is \( A(n)=A(1)+d(n - 1) \). Substituting \( A(1) = 21 \) and \( d = 4 \), we get:
\[

$$\begin{align*} A(n)&=21 + 4(n - 1)\\ &=21+4n - 4\\ &=17 + 4n \end{align*}$$

\]

Step3: Find the 21st term

Substitute \( n = 21 \) into \( A(n)=17 + 4n \):
\[
A(21)=17+4\times21=17 + 84 = 101
\]

Answer:

D. \( A(n)=17 + 4n \); \( A(21)=101 \)