Subject Test Math 2

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Subject Test Math 2

Question 23 of 28

For some real number $t$ , the first three terms of an arithmetic sequence are $2 t comma space$ $5 t minus 1 comma space$ and $6 t + 2$ . What is the numerical value of the fourth term?

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$4$

Correct Answer:

No

$8$

Correct Answer:

No

$10$

Correct Answer:

No

$16$

Correct Answer:

No

$19$

Correct Answer:

Yes

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The correct answer is E.

To determine the numerical value of the fourth term, first determine the value of $t$ and then apply the common difference.

Since $2 t, 5 t - 1$ , and $6 t + 2$ are the first three terms of an arithmetic sequence, it must be true that $(6 t + 2) - (5 t - 1) = (5 t - 1) - 2 t$ , that is, $t + 3 = 3 t - 1.$ Solving $t + 3 = 3 t - 1$ for $t$ gives $t = 2$ . Substituting $2$ for $t$ in the expressions of the three first terms of the sequence, one sees that they are $4$ , $9$ , and $14$ , respectively. The common difference between consecutive terms for this arithmetic sequence is $5 = 14 - 9 = 9 - 4$ , and therefore, the fourth term is $14 + 5 = 19$ .