Question
Describe the pattern, write the next term, and write a rule for the \( n \)th term of the sequence. \( 1,2,4,8, \ldots \) The pattern is The next term is The \( n \)th term is \( a_{n}= \)
Ask by Rowe Pierce. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**The pattern is:**
Each term is multiplied by 2 to get the next term.
**The next term is:**
16
**The \( n \)th term is:**
\( a_{n} = 2^{n-1} \)
Solution
**The pattern is:**
Each term is obtained by multiplying the previous term by \( 2 \).
**The next term is:**
\( 16 \)
(As \( 8 \times 2 = 16 \))
**The \( n \)th term is:**
\( a_{n} = 2^{n-1} \)
(This formula works for \( n = 1, 2, 3, \ldots \) as \( a_1 = 2^{0} = 1 \), \( a_2 = 2^{1} = 2 \), etc.)
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The pattern is that each term in the sequence is obtained by multiplying the previous term by 2. It starts with 1, and then each successive term doubles the one before it: \( 1 \times 2 = 2 \), \( 2 \times 2 = 4 \), \( 4 \times 2 = 8 \), and so on. The next term is \( 16 \) (since \( 8 \times 2 = 16 \)). The \( n \)th term is \( a_{n} = 2^{(n-1)} \). This means for any term number \( n \), you can find the term by raising 2 to the power of \( n-1 \). So, for example, when \( n = 5 \), you're looking for \( 2^{4} = 16 \)!
