1.
Find the sum of the first \(50\) terms of the following arithmetic series: \(6+8+10+12+\cdots\)
Step 1. Identify the first term, common difference, and number of terms. \[ a=6,\quad d=8-6=2,\quad n=50 \] Step 2. Use the arithmetic series sum formula. \[ S_n=\frac{n}{2}\left(2a+(n-1)d\right) \] Step 3. Substitute the values. \[ S_{50}=\frac{50}{2}\left(2(6)+(50-1)(2)\right) \] Step 4. Simplify. \[ \begin{aligned} S_{50}&=25(12+98)\\&=25(110)\\&=2750\end{aligned} \] Answer: A
2.
Determine the sum of the arithmetic series: \(4+11+18+\cdots+88\)
Step 1. Identify the first term, common difference, and last term. \[ a=4,\quad d=11-4=7,\quad t_n=88 \] Step 2. Find the number of terms using the general term formula. \[ t_n=a+(n-1)d \] Step 3. Substitute the values. \[ 88=4+(n-1)(7) \] Step 4. Solve for \(n\). \[ \begin{aligned} 88&=4+7n-7\\88&=7n-3\\91&=7n\\13&=n\end{aligned} \] Step 5. Use the arithmetic series sum formula with first and last terms. \[ S_n=\frac{n(a+l)}{2} \] Step 6. Substitute \(n=13\), \(a=4\), and \(l=88\). \[ S_{13}=\frac{13(4+88)}{2}=598 \] Answer: C
3.
In an arithmetic sequence, which of these represents the fourteenth term?
Step 1. In a sequence, \(a_n\) represents the \(n\)th term. Step 2. The fourteenth term means \(n=14\). Step 3. Therefore, the fourteenth term is written as: \[ a_{14} \] So the correct representation is \(a_{14}\). Answer: C
4.
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
5.
Which of the following is NOT an arithmetic series?
Step 1. An arithmetic series has a constant common difference. Step 2. Check each series. For \(1+8+15+22+\cdots\): \[ 8-1=7,\quad 15-8=7,\quad 22-15=7 \] This is arithmetic. For \(-22-13-4+5+\cdots\): \[ -13-(-22)=9,\quad -4-(-13)=9,\quad 5-(-4)=9 \] This is arithmetic. For \(-22.5-27-31.5-36-\cdots\): \[ -27-(-22.5)=-4.5,\quad -31.5-(-27)=-4.5,\quad -36-(-31.5)=-4.5 \] This is arithmetic. For \(93.3+87.2+81.1+75+\cdots\): \[ 87.2-93.3=-6.1,\quad 81.1-87.2=-6.1,\quad 75-81.1=-6.1 \] This is arithmetic. Step 3. Since all listed series are arithmetic, the correct choice is that these are all arithmetic series. Answer: C
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