Find an explicit formula for a sequence of the form a, a, a, 4, 8, 16, 32, 64, 128, 256 I an = with the initial terms given below.
Find an explicit formula for a sequence of the form a, a, a, 4, 8, 16, 32, 64, 128, 256 I an = with the initial terms given below.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Find an explicit formula for a sequence of the form \( a_1, \ a_2, \ a_3, \ \ldots \) with the initial terms given below.
\[ 4, 8, 16, 32, 64, 128, 256 \]
\[ a_n = \]
### Explanation
This sequence is a geometric sequence. Each term is obtained by multiplying the previous term by a constant factor. Here, the factor is 2, starting from the first term, 4. The sequence represents powers of 2 multiplied by 4:
- \( a_1 = 4 \times 2^0 = 4 \)
- \( a_2 = 4 \times 2^1 = 8 \)
- \( a_3 = 4 \times 2^2 = 16 \)
- \( a_4 = 4 \times 2^3 = 32 \)
- \( \ldots \)
Thus, the explicit formula for the sequence is:
\[ a_n = 4 \times 2^{n-1} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F1de25e5d-6612-444b-bbd4-6aa52baeea5e%2F9vr06ko_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Find an explicit formula for a sequence of the form \( a_1, \ a_2, \ a_3, \ \ldots \) with the initial terms given below.
\[ 4, 8, 16, 32, 64, 128, 256 \]
\[ a_n = \]
### Explanation
This sequence is a geometric sequence. Each term is obtained by multiplying the previous term by a constant factor. Here, the factor is 2, starting from the first term, 4. The sequence represents powers of 2 multiplied by 4:
- \( a_1 = 4 \times 2^0 = 4 \)
- \( a_2 = 4 \times 2^1 = 8 \)
- \( a_3 = 4 \times 2^2 = 16 \)
- \( a_4 = 4 \times 2^3 = 32 \)
- \( \ldots \)
Thus, the explicit formula for the sequence is:
\[ a_n = 4 \times 2^{n-1} \]
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