Pregunta
\( 6,9,12,15, \ldots \) (a) Find the common difference. The common difference is \( d=3 \). (Type a whole number.) (b) Find the eighth term. The eighth term is \( a_{3}=\square \). (Type a whole number.)
Ask by Murphy May. in the United States
Jan 23,2025
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(a) The common difference is 3.
(b) The eighth term is 27.
Solución
### (a) Find the common difference.
The common difference \( d \) is calculated by subtracting the first term from the second term:
\[
d = 9 - 6 = 3
\]
**Answer:** \( 3 \)
---
### (b) Find the eighth term.
To find the eighth term \( a_8 \) of an arithmetic sequence, use the formula:
\[
a_n = a_1 + (n-1)d
\]
Where:
- \( a_1 = 6 \) (the first term)
- \( d = 3 \) (the common difference)
- \( n = 8 \) (the term number)
Plugging in the values:
\[
a_8 = 6 + (8-1) \times 3 = 6 + 7 \times 3 = 6 + 21 = 27
\]
**Answer:** \( 27 \)
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Extra Insights
To find the eighth term, we can use the formula for the \( n \)-th term of an arithmetic sequence, which is given by: \[ a_n = a_1 + (n-1)d \] In this case, the first term \( a_1 = 6 \) and the common difference \( d = 3 \). Plugging in the values for the eighth term \( n=8 \): \[ a_8 = 6 + (8-1) \cdot 3 = 6 + 21 = 27 \] So, the eighth term is \( a_8 = 27 \).
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