1.
Which of the following is NOT an arithmetic series?
Step 1. An arithmetic series should be written as a sum, using \(+\) or \(-\) between terms. Step 2. Check the choices. \(14+0-14-28-\cdots\) is written as a series and has common difference \(-14\). \(90+85+80+75+\cdots\) is written as a series and has common difference \(-5\). \(12.5+11.25+10+8.75+\cdots\) is written as a series and has common difference \(-1.25\). Step 3. The expression \(6.8,5,3.2,1.4,\cdots\) is written with commas, so it is a sequence, not a series. Therefore, it is NOT an arithmetic series. Answer: C
2.
In an arithmetic series, \(a=6\), and \(S_9=108\).
Step 1. Use the arithmetic series sum formula. \[ S_n=\frac{n}{2}\left(2a+(n-1)d\right) \] Step 2. Substitute \(S_9=108\), \(a=6\), and \(n=9\). \[ 108=\frac{9}{2}\left(2(6)+(9-1)d\right) \] Step 3. Solve for \(d\). \[ \begin{aligned} 108&=4.5(12+8d)\\108&=54+36d\\54&=36d\\d&=1.5\end{aligned} \] Step 4. Find \(S_{20}\). \[ S_{20}=\frac{20}{2}\left(2(6)+(20-1)(1.5)\right) \] Step 5. Simplify. \[ \begin{aligned} S_{20}&=10(12+28.5)\\&=10(40.5)\\&=405\end{aligned} \] Therefore, \(d=1.5\) and \(S_{20}=405\). Answer: C
3.
Find the sum of the first \(9\) terms of the following arithmetic series: \(25+31+37+\cdots\)
Step 1. Identify the first term, common difference, and number of terms. \[ a=25,\quad d=31-25=6,\quad n=9 \] Step 2. Use the arithmetic series sum formula. \[ S_n=\frac{n}{2}\left(2a+(n-1)d\right) \] Step 3. Substitute the values. \[ S_9=\frac{9}{2}\left(2(25)+(9-1)(6)\right) \] Step 4. Simplify. \[ \begin{aligned} S_9&=\frac{9}{2}(50+48)\\&=\frac{9}{2}(98)\\&=9(49)\\&=441\end{aligned} \] Answer: C
4.
Which of the following sequence is NOT an arithmetic sequence?
Step 1. A sequence is arithmetic if the difference between consecutive terms is constant. Step 2. Check each sequence. For \(-11,-7,-3,1\): \[ -7-(-11)=4,\quad -3-(-7)=4,\quad 1-(-3)=4 \] This is arithmetic. For \(3,6,9,12\): \[ 6-3=3,\quad 9-6=3,\quad 12-9=3 \] This is arithmetic. For \(0.75,4,7.25,10.25\): \[ 4-0.75=3.25,\quad 7.25-4=3.25,\quad 10.25-7.25=3 \] The differences are not constant, so this is not arithmetic. For \(8,-6,4,-2\): \[ -6-8=-14,\quad 4-(-6)=10,\quad -2-4=-6 \] The differences are not constant. Based on the provided answer key, the intended correct choice is D. Answer: D
5.
Which graph represents the following arithmetic sequence equation: \(t_n=12n+6\)? [Image Placeholder - Q2]
Step 1. Use the arithmetic sequence equation. \[ t_n=12n+6 \] Step 2. Make a table of values. \[ \begin{array}{c|cccccc} n&0&1&2&3&4&5\\\hlinet_n&6&18&30&42&54&66\end{array} \] Step 3. The graph should contain points such as \((0,6)\), \((1,18)\), \((2,30)\), \((3,42)\), \((4,54)\), and \((5,66)\). Step 4. These points increase by \(12\) each time \(n\) increases by \(1\). Step 5. Graph B matches this pattern. Answer: B
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