Question
Find the number of terms in the following arithmetic progression: \( 3,9,15,21, \ldots, 87 \) A. 9 B. 12 C. 18 D. 15
Ask by Owen Gough. in the United States
Mar 03,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
There are 15 terms in the arithmetic progression.
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Solution
The arithmetic progression is 3, 9, 15, 21, …, 87.
Step 1: Determine the first term (a₁) and the common difference (d).
a₁ = 3
d = 9 − 3 = 6
Step 2: Use the formula for the nth term of an arithmetic progression:
aₙ = a₁ + (n − 1) * d
Step 3: Substitute the known values and set aₙ = 87:
87 = 3 + (n − 1) * 6
Step 4: Solve for n:
87 − 3 = (n − 1) * 6
84 = (n − 1) * 6
n − 1 = 84 / 6
n − 1 = 14
n = 14 + 1
n = 15
Thus, there are 15 terms in the progression. The correct answer is D. 15.
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Simplify this solution Bonus Knowledge
To find the number of terms in the arithmetic progression \( 3, 9, 15, 21, \ldots, 87 \), we first identify the first term \( a \) and the common difference \( d \). The first term: \[ a = 3 \] The common difference: \[ d = 9 - 3 = 6 \] The general formula for the \( n \)-th term of an arithmetic progression is: \[ a_n = a + (n-1)d \] Setting \( a_n = 87 \) (the last term), we can solve for \( n \): \[ 87 = 3 + (n-1)6 \] Subtract \( 3 \) from both sides: \[ 84 = (n-1)6 \] Now, divide by \( 6 \): \[ 14 = n - 1 \] Finally, add \( 1 \) to find \( n \): \[ n = 15 \] Thus, the number of terms in the arithmetic progression is \( \boxed{15} \).
