The first term is \(\displaystyle x_{0} = 1,000\).

The common difference is

 \(\displaystyle 997 - 1,000 = -3\).

The seventieth term is 

\(\displaystyle x_{69} = x_{0}+ 69 \cdot d= 1,000 + 69 \cdot (-3)= 1,000 - 207 = 793\).

The first term is \(\displaystyle x_{0} = 4\); the common difference is

\(\displaystyle d = 9-4 = 5\).

The hundredth term is 

\(\displaystyle x_{99} = x_{0} + 99 d = 4 + 99 \cdot5 = 4 + 495 = 499\).

The common difference of the sequence is

\(\displaystyle -124.7 - (-128) = 3.3\),

so the \(\displaystyle n\)th term of the sequence is

\(\displaystyle a_{n} = -128 + 3.3 (n-1)\)

To find out the minimum value for which \(\displaystyle a_{n} > 0\), set up this inequality:

\(\displaystyle -128 + 3.3 (n-1) > 0\)

\(\displaystyle -128 + 3.3 n-3.3 > 0\)

\(\displaystyle 3.3 n-131.3 > 0\)

\(\displaystyle 3.3 n>131.3\)

\(\displaystyle n > 39.8\)

The first positive term is the fortieth term.

The common difference of the sequence is

\(\displaystyle 8.1 - 5.7 = 2.4\)

so the \(\displaystyle n\)th term of the sequence is

\(\displaystyle a_{n} =5.7+2.4 (n-1)\)

To find out the minimum value for which \(\displaystyle a_{n} > 100\), set up this inequality:

\(\displaystyle 5.7+2.4 (n-1) > 100\)

\(\displaystyle 5.7+2.4 n-2.4 > 100\)

\(\displaystyle 2.4 n +3.3 > 100\)

\(\displaystyle 2.4 n > 96.7\)

\(\displaystyle n > 96.7 \div 2.4 \approx 40.3\)

The forty-first term is the correct response.

The common difference of the sequence is

\(\displaystyle 297.3 - 300 = -2.7\)

so the \(\displaystyle n\)th term of the sequence is

\(\displaystyle a_{n} = 300 -2.7 (n-1)\)

To find out the minimum value for which \(\displaystyle a_{n} < 0\), set up this inequality:

\(\displaystyle 300 -2.7 (n-1) < 0\)

\(\displaystyle 300 -2.7 n+2.7 < 0\)

\(\displaystyle 302.7-2.7 n < 0\)

\(\displaystyle 302.7 < 2.7 n\)

\(\displaystyle 2.7 n > 302.7\)

\(\displaystyle n > 302.7 \div 2.7 \approx 112.1\)

The first negative term is the one hundred thirteenth term.

The common difference of the sequence is

\(\displaystyle 153 \frac{3}{4}- 155 \frac{5}{6}= -2 \frac{1}{12}\)

so the \(\displaystyle n\)th term of the sequence is

\(\displaystyle a_{n} = 155\frac{5}{6}-\left (2 \frac{1}{12} \right )(n-1)\)

To find out the minimum value for which \(\displaystyle a_{n} < 0\), set up this inequality:

\(\displaystyle 155\frac{5}{6}-\left (2 \frac{1}{12} \right )(n-1) < 0\)

\(\displaystyle 155\frac{5}{6}-\left (2 \frac{1}{12} \right ) n + 2\frac{1}{12} < 0\)

\(\displaystyle 157\frac{11}{12}-\left (2 \frac{1}{12} \right ) n < 0\)

\(\displaystyle 157\frac{11}{12} < \left (2 \frac{1}{12} \right ) n\)

\(\displaystyle \left (2 \frac{1}{12} \right ) n > 157\frac{11}{12}\)

\(\displaystyle n > 157\frac{11}{12} \div 2 \frac{1}{12} = 75 \frac{4}{5}\)

The seventy-sixth term is the first negative term.

The common difference of the sequence is

\(\displaystyle -75 \frac{4}{5} - \left ( -78 \frac{2}{3} \right ) =2 \frac{13}{15}\),

so the \(\displaystyle n\)th term of the sequence is

\(\displaystyle a_{n} = - 78\frac{2}{3} + \left ( 2\frac{13}{15} \right ) (n-1)\)

To find out the minimum value for which \(\displaystyle a_{n} > 0\), set up this inequality:

\(\displaystyle - 78\frac{2}{3} + \left ( 2\frac{13}{15} \right ) (n-1) > 0\)

\(\displaystyle - 78\frac{2}{3} + \left ( 2\frac{13}{15} \right ) n - 2\frac{13}{15} > 0\)

\(\displaystyle \left ( 2\frac{13}{15} \right ) n - 81\frac{8}{15} > 0\)

\(\displaystyle \left ( 2\frac{13}{15} \right ) n > 81\frac{8}{15}\)

\(\displaystyle n > 81\frac{8}{15} \div 2\frac{13}{15} = 28 \frac{19}{43}\)

The first positive term in the sequence is the twenty-ninth term.

The common difference of the sequence is

\(\displaystyle 3 \frac{1}{4} - 1 \frac{1}{7} = 2\frac{3}{28}\)

so the \(\displaystyle n\)th term of the sequence is

\(\displaystyle a_{n} = 1\frac{1}{7} + \left ( 2\frac{3}{28} \right ) (n-1)\)

To find out the minimum value for which \(\displaystyle a_{n} > 100\), set up this inequality:

\(\displaystyle 1\frac{1}{7} + \left ( 2\frac{3}{28} \right ) (n-1) > 100\)

\(\displaystyle 1\frac{1}{7} + \left ( 2\frac{3}{28} \right ) n -2\frac{3}{28} > 100\)

\(\displaystyle \left ( 2\frac{3}{28} \right ) n - \frac{27}{28} > 100\)

\(\displaystyle \left ( 2\frac{3}{28} \right ) n > 100 \frac{27}{28}\)

\(\displaystyle n > 100 \frac{27}{28} \div 2\frac{3}{28} = 47 \frac{54}{59}\)

The correct response is the forty-eighth term.

The \(\displaystyle n\)th term of an arithmetic sequence with initial term \(\displaystyle a_{1}\) and common difference \(\displaystyle d\) is defined by the equation

\(\displaystyle a_{n} = a_{1}+ (n-1)d\)

Since the tenth and twelfth terms are two terms apart, the common difference can be found as follows:

\(\displaystyle a_{10}+2d=a_{12}\)

\(\displaystyle 8.4+2d=10.2\)

\(\displaystyle 8.4+2d- 8.4=10.2 - 8.4\)

\(\displaystyle 2d=1.8\)

\(\displaystyle d = 0.9\)

Now, we can set \(\displaystyle n = 10, d = 0.9\) in the sequence equation to find \(\displaystyle a_{1}\):

\(\displaystyle a_{n} = a_{1}+ (n-1)d\)

\(\displaystyle a_{10} = a_{1}+ (10-1)0.9\)

\(\displaystyle 8.4 = a_{1}+ 9 \cdot 0.9\)

\(\displaystyle 8.4 = a_{1}+ 8.1\)

\(\displaystyle 8.4 - 8.1 = a_{1}+ 8.1 - 8.1\)

\(\displaystyle a_{1} = 0.3\)

The \(\displaystyle n\)th term of an arithmetic sequence with initial term \(\displaystyle a_{1}\) and common difference \(\displaystyle d\) is defined by the equation

\(\displaystyle a_{n} = a_{1}+ (n-1)d\)

Since the eleventh and thirteenth terms are two terms apart, the common difference can be found as follows:

\(\displaystyle a_{11}+2d=a_{13}\)

\(\displaystyle 11+2d=14\)

\(\displaystyle 11+2d -11=14 -11\)

\(\displaystyle 2d = 3\)

\(\displaystyle d = \frac{3}{2}\)

Now, we can set \(\displaystyle n = 11, d =\frac{3}{2}\) in the sequence equation to find \(\displaystyle a_{1}\):

\(\displaystyle a_{11} = a_{1}+ (11-1) \frac{3}{2}\)

\(\displaystyle 11= a_{1}+ 10 \cdot \frac{3}{2}\)

\(\displaystyle 11= a_{1}+ 15\)

\(\displaystyle 11- 15= a_{1}+ 15 - 15\)

\(\displaystyle a_{1} = -4\)