Question
Find the \( 70^{\text {th }} \) term of the following arithmetic sequence. \[ 12,20,28,36, \ldots \]
Ask by Cox Wagner. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The 70th term of the sequence is 564.
Solution
1. The general formula for the \( n^{\text{th}} \) term of an arithmetic sequence is
\[
a_n = a_1 + (n-1)d,
\]
where \( a_1 \) is the first term and \( d \) is the common difference.
2. From the sequence \( 12, \, 20, \, 28, \, 36, \ldots \), we identify
\[
a_1 = 12 \quad \text{and} \quad d = 20 - 12 = 8.
\]
3. To find the \( 70^{\text{th}} \) term, substitute \( n = 70 \) into the formula:
\[
a_{70} = 12 + (70-1) \times 8 = 12 + 69 \times 8.
\]
4. Compute the multiplication:
\[
69 \times 8 = 552.
\]
5. Now, add the value to the first term:
\[
12 + 552 = 564.
\]
Thus, the \( 70^{\text{th}} \) term of the sequence is
\[
\boxed{564}.
\]
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Simplify this solution The Deep Dive
To find the \( 70^{\text{th}} \) term of the arithmetic sequence \( 12, 20, 28, 36, \ldots \), we first need to identify the first term \( a \) and the common difference \( d \). The first term \( a \) is \( 12 \). The common difference \( d \) can be calculated by subtracting the first term from the second term: \[ d = 20 - 12 = 8. \] The formula for the \( n \)-th term of an arithmetic sequence is given by: \[ a_n = a + (n-1)d. \] Substituting the known values, where \( n = 70 \): \[ a_{70} = 12 + (70-1) \cdot 8. \] \[ a_{70} = 12 + 69 \cdot 8. \] \[ a_{70} = 12 + 552. \] \[ a_{70} = 564. \] Therefore, the \( 70^{\text{th}} \) term of the sequence is \( \boxed{564} \).
