Question

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an arithmetic sequence has 14 as its third term and 26 as its sixth term. what is the function that represents the sequence?
f(n) = 14 + 12(n - 1)
f(n) = 6 + 4(n - 1)
f(n) = 6 + 12(n - 1)
f(n) = 14 + 4(n - 1)

Explanation:

Step1: Recall arithmetic sequence formula

The nth term of an arithmetic sequence is given by $f(n) = a_1 + d(n-1)$, where $a_1$ is the first term and $d$ is the common difference.

Step2: Set up equations for given terms

For the 3rd term: $14 = a_1 + d(3-1) \implies 14 = a_1 + 2d$
For the 6th term: $26 = a_1 + d(6-1) \implies 26 = a_1 + 5d$

Step3: Solve for common difference $d$

Subtract the first equation from the second:
$26 - 14 = (a_1 + 5d) - (a_1 + 2d)$
$12 = 3d \implies d = 4$

Step4: Solve for first term $a_1$

Substitute $d=4$ into $14 = a_1 + 2d$:
$14 = a_1 + 2(4) \implies 14 = a_1 + 8 \implies a_1 = 6$

Step5: Write the sequence function

Substitute $a_1=6$ and $d=4$ into the nth term formula: $f(n) = 6 + 4(n-1)$

Answer:

B. $f(n)=6 + 4(n - 1)$