Question
An arithmetic sequence has a first term of 10 and a sixth term of 40. What is the 20 th term of this sequence? A. 105 B. 110 C. 124 D. 130
Ask by Clark Cummings. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The 20th term of the sequence is 124.
Solution
Let the arithmetic sequence be defined by the formula
\[
a_n = a_1 + (n-1)d,
\]
where \( a_1 \) is the first term and \( d \) is the common difference.
Given:
\[
a_1 = 10 \quad \text{and} \quad a_6 = 40.
\]
Substitute \( n = 6 \) into the formula:
\[
40 = 10 + 5d.
\]
Solve for \( d \):
\[
5d = 40 - 10 = 30,
\]
\[
d = \frac{30}{5} = 6.
\]
Now, use the formula for the 20th term:
\[
a_{20} = 10 + 19 \times 6.
\]
Calculate:
\[
a_{20} = 10 + 114 = 124.
\]
Thus, the 20th term is \( \boxed{124} \).
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Simplify this solution Bonus Knowledge
To find the 20th term of the arithmetic sequence, we first need to determine the common difference. The first term \( a_1 \) is 10 and the sixth term \( a_6 \) can be represented as \( a_6 = a_1 + 5d \) where \( d \) is the common difference. So, \[ 40 = 10 + 5d \] Subtracting 10 from both sides, we get: \[ 30 = 5d \] Dividing both sides by 5 gives us: \[ d = 6 \] Now, we can find the 20th term using the formula for the \( n \)-th term of an arithmetic sequence, \( a_n = a_1 + (n-1)d \): \[ a_{20} = 10 + (20-1) \times 6 = 10 + 114 = 124 \] So, the 20th term is \( \boxed{124} \).
