1.
Find the sum of the first \(60\) terms of the following arithmetic series: \(44+41+38+35+\cdots\)
Step 1. Identify the first term, common difference, and number of terms. \[ a=44,\quad d=41-44=-3,\quad n=60 \] Step 2. Use the arithmetic series sum formula. \[ S_n=\frac{n}{2}\left(2a+(n-1)d\right) \] Step 3. Substitute the values. \[ S_{60}=\frac{60}{2}\left(2(44)+(60-1)(-3)\right) \] Step 4. Simplify. \[ \begin{aligned} S_{60}&=30\left(88+59(-3)\right)\\&=30(88-177)\\&=30(-89)\\&=-2670\end{aligned} \] Answer: A
2.
In an arithmetic sequence, which of these represents the fourteenth term?
Step 1. In a sequence, the notation \(a_n\) represents the \(n\)th term. Step 2. The fourteenth term means \(n=14\). Step 3. Substitute \(n=14\) into \(a_n\). \[ a_n=a_{14} \] Therefore, the fourteenth term is represented by \(a_{14}\). Answer: E
3.
Determine the sum of the arithmetic series: \(2+7+12+\cdots+62\)
Step 1. Identify the first term, common difference, and last term. \[ a=2,\quad d=7-2=5,\quad t_n=62 \] Step 2. Use the general term formula to find \(n\). \[ t_n=a+(n-1)d \] Step 3. Substitute the values. \[ 62=2+(n-1)(5) \] Step 4. Solve for \(n\). \[ \begin{aligned} 62&=2+5n-5\\62&=5n-3\\65&=5n\\13&=n\end{aligned} \] Step 5. Use the arithmetic series sum formula with first and last terms. \[ S_n=\frac{n(a+l)}{2} \] Step 6. Substitute \(n=13\), \(a=2\), and \(l=62\). \[ S_{13}=\frac{13(2+62)}{2}=416 \] Answer: A
4.
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 \(1+5+9+13+\cdots\): \[ 5-1=4,\quad 9-5=4,\quad 13-9=4 \] This is arithmetic. For \(21+32+43+54+\cdots\): \[ 32-21=11,\quad 43-32=11,\quad 54-43=11 \] This is arithmetic. For \((-4)+(-1)+2+5+\cdots\): \[ (-1)-(-4)=3,\quad 2-(-1)=3,\quad 5-2=3 \] This is arithmetic. For \(3+7+12+18+\cdots\): \[ 7-3=4,\quad 12-7=5,\quad 18-12=6 \] The common difference is not constant, so this is not an arithmetic series. Answer: E
5.
Which graph represents the following arithmetic sequence equation: \(t_n=4n+6\)? [Image Placeholder - Q2]
Step 1. Use the arithmetic sequence equation. \[ t_n=4n+6 \] Step 2. Find several ordered pairs \((n,t_n)\). \[ \begin{aligned} t_1&=4(1)+6=10\\t_2&=4(2)+6=14\\t_3&=4(3)+6=18\\t_4&=4(4)+6=22\\t_5&=4(5)+6=26\end{aligned} \] Step 3. The graph should contain the points \((1,10),(2,14),(3,18),(4,22),(5,26)\). Step 4. Graph C shows these increasing points with common difference \(4\). Answer: C
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