A. What is the sum of theterms of each finite sequence.1. 13, 16, 19, 22, 25, 28, 312. 17, 27, 37, 47,573. -1, -9, -17, -25, -33,-414. 125, 130, 135, 140, 1455.-9, -5, -1, 3, 7, 11, 15What answer this question?
Answer (1)
1. SOLUTION:Step 1: List the given values.[tex]\begin{aligned} & n = 7 \\ & a_1 = 13 \\ & a_2 = 16 \\ & a_3 = 19 \\ & a_7 = 31 \end{aligned}[/tex]Step 2: Find the common difference.[tex]\begin{aligned} d & = a_2 - a_1 = a_3 - a_2 \\ d & = 16 - 13 = 19 - 16 \\ d & = 3 = 3 \end{aligned}[/tex]Step 3: Solve for the sum of the terms of an arithmetic sequence.[tex]\begin{aligned} S & = \frac{n}{2}[2a_1 + (n - 1)d] \\ & = \frac{7}{2}[2(13) + (7 - 1)(3)] \\ & = 3.5[(26 + (6)(3)] \\ & = 3.5(26 + 18) \\ & = 3.5(44) \\ & = \boxed{154} \end{aligned}[/tex]2. SOLUTION:Step 1: List the given values.[tex]\begin{aligned} & n = 5 \\ & a_1 = 17 \\ & a_2 = 27 \\ & a_3 = 37 \\ & a_5 = 57 \end{aligned}[/tex]Step 2: Find the common difference.[tex]\begin{aligned} d & = a_2 - a_1 = a_3 - a_2 \\ d & = 27 - 17 = 37 - 27 \\ d & = 10 = 10 \end{aligned}[/tex]Step 3: Solve for the sum of the terms of an arithmetic sequence.[tex]\begin{aligned} S & = \frac{n}{2}[2a_1 + (n - 1)d] \\ & = \frac{5}{2}[2(17) + (5 - 1)(10)] \\ & = 2.5[(34 + (4)(10)] \\ & = 2.5(34 + 40) \\ & = 2.5(74) \\ & = \boxed{185} \end{aligned}[/tex]3. SOLUTION:Step 1: List the given values.[tex]\begin{aligned} & n = 6 \\ & a_1 = -1 \\ & a_2 = -9 \\ & a_3 = -17 \\ & a_6 = -41 \end{aligned}[/tex]Step 2: Find the common difference.[tex]\begin{aligned} d & = a_2 - a_1 = a_3 - a_2 \\ d & = -9 - (-1) = -17 - (-9) \\ d & = -9 + 1 = -17 + 9 \\ d & = -8 = -8 \end{aligned}[/tex]Step 3: Solve for the sum of the terms of an arithmetic sequence.[tex]\begin{aligned} S & = \frac{n}{2}[2a_1 + (n - 1)d] \\ & = \frac{6}{2}[2(-1) + (6 - 1)(-8)] \\ & = 3[(-2+ (5)(-8)] \\ & = 3(-2 + (-40)) \\ & = 3(-42) \\ & = \boxed{-126} \end{aligned}[/tex]4. SOLUTION:Step 1: List the given values.[tex]\begin{aligned} & n = 5 \\ & a_1 = 125 \\ & a_2 = 130 \\ & a_3 = 135 \\ & a_5 = 145 \end{aligned}[/tex]Step 2: Find the common difference.[tex]\begin{aligned} d & = a_2 - a_1 = a_3 - a_2 \\ d & = 130 - 125 = 135 - 130 \\ d & = 5 = 5 \end{aligned}[/tex]Step 3: Solve for the sum of the terms of an arithmetic sequence.[tex]\begin{aligned} S & = \frac{n}{2}[2a_1 + (n - 1)d] \\ & = \frac{5}{2}[2(125) + (5 - 1)(5)] \\ & = 2.5[(250 + (4)(5)] \\ & = 2.5(250 + 20) \\ & = 2.5(270) \\ & = \boxed{675} \end{aligned}[/tex]5. SOLUTION:Step 1: List the given values.[tex]\begin{aligned} & n = 7 \\ & a_1 = -9 \\ & a_2 = -5 \\ & a_3 = -1 \\ & a_7 = 15 \end{aligned}[/tex]Step 2: Find the common difference.[tex]\begin{aligned} d & = a_2 - a_1 = a_3 - a_2 \\ d & = -5 - (-9) = -1 - (-5) \\ d & = -5 + 9 = -1 + 5 \\ d & = 4 = 4 \end{aligned}[/tex]Step 3: Solve for the sum of the terms of an arithmetic sequence.[tex]\begin{aligned} S & = \frac{n}{2}[2a_1 + (n - 1)d] \\ & = \frac{7}{2}[2(-9) + (7 - 1)(4)] \\ & = 3.5[(-18 + (6)(4)] \\ & = 3.5(-18 + 24) \\ & = 3.5(6) \\ & = \boxed{21} \end{aligned}[/tex]
