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Numbers and Operations

SAT Sequences and Series

\(1, -8, t, -62, -134, \ldots\)

In the sequence above, the first term is \(1\). Each term after the first is \(10\) less than \(2\) times the previous term. What is the value of \(t\)?

(A) \(\ \ -278\)
(B) \(\ \ -26\)
(C) \(\ \ -18\)
(D) \(\ \ 8\)
(E) \(\ \ 10\)

\(M=12341234\ldots 1234\)

\(M\) is formed by writing \(1234\) repeatedly, as shown above. If \(M\) has \(128\) digits, which of the following is the sum of its digits?

(A) \(\ \ 10\)
(B) \(\ \ 32\)
(C) \(\ \ 128\)
(D) \(\ \ 320\)
(E) \(\ \ 3200\)

In the decimal \(0.9756497564\ldots\), the digits \(97564\) repeat indefinitely. Which digit is in the \(4192\)th place to the right of the decimal point?

(A) \(\ \ 9\)
(B) \(\ \ 7\)
(C) \(\ \ 6\)
(D) \(\ \ 5\)
(E) \(\ \ 4\)

Consider the arithmetic sequence \(1, \frac{1}{2}, 0, -\frac{1}{2}, \ldots\). What is the \(8\)th term in the sequence?

(A) \(\ \ -3\)
(B) \(\ -\frac{5}{2}\)
(C) \(\ \ -2\)
(D) \(\ \ \frac{5}{2}\)
(E) \(\ \ 8\)

\(28, 7, \frac{7}{4}, \frac{7}{16}, \ldots\)

In the sequence above, each term is obtained by multiplying the preceding term by \(\frac{1}{4}\). What is the \(91\)th term of the sequence?

(A) \(\ \ (\frac{1}{4})^{89}\)
(B) \(\ \ 7 (\frac{1}{4})^{89}\)
(C) \(\ \ 28 (\frac{1}{4})^{89}\)
(D) \(\ \ 7(\frac{1}{4})^{90}\)
(E) \(\ \ 7(\frac{1}{4})^{91}\)

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