23.3: Exercises
- Page ID
- 54478
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Find the first seven terms of the sequence.
- \(a_n=3n\)
- \(a_n=5n+3\)
- \(a_n=n^2+2\)
- \(a_n=n\)
- \(a_n=(-1)^{n+1}\)
- \(a_n=\dfrac{\sqrt{n+1}}{n}\)
- \(a_k=10^k\)
- \(a_i=5+(-1)^i\)
- Answer
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- \(3, 6, 9, 12, 15, 18, 21\)
- \(8, 13, 18, 23, 28, 33, 38\)
- \(3, 6, 11, 18, 27, 38, 51\)
- \(1, 2, 3, 4, 5, 6, 7\)
- \(1, −1, 1, −1, 1, −1, 1\)
- \(\sqrt{2}, \dfrac{\sqrt{3}}{2}, \dfrac{2}{3}, \dfrac{\sqrt{5}}{4}, \dfrac{\sqrt{6}}{5}, \dfrac{\sqrt{7}}{6}, \dfrac{\sqrt{8}}{7}\)
- \(10, 100, 1000, 10000, 100000, 1000000, 10000000\)
- \(4, 6, 4, 6, 4, 6, 4\)
Find the first six terms of the sequence.
- \(a_1=5\), \(a_n=a_{n-1}+3\) for \(n\geq 2\)
- \(a_1=7\), \(a_n=10\cdot a_{n-1}\) for \(n\geq 2\)
- \(a_1=1\), \(a_n=2\cdot a_{n-1}+1\) for \(n\geq 2\)
- \(a_1=6\), \(a_2=4\), \(a_n=a_{n-1}-a_{n-2}\) for \(n\geq 3\)
- Answer
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- \(5, 8, 11, 14, 17\)
- \(7, 70, 700, 7000, 70000\)
- \(1, 3, 7, 15, 31\)
- \(6, 4, −2, −6, −4\)
Find the value of the series.
- \(\sum_{n=1}^4 a_n\), where \(a_n=5n\)
- \(\sum_{k=1}^5 a_k\), where \(a_k=k\)
- \(\sum_{i=1}^4 a_i\), where \(a_n=n^2\)
- \(\sum_{n=1}^6 (n-4)\)
- \(\sum_{k=1}^3 (k^2+4k-4)\)
- \(\sum_{j=1}^4 \dfrac{1}{j+1}\)
- Answer
-
- \(50\)
- \(15\)
- \(30\)
- \(−3\)
- \(26\)
- \(\dfrac {77}{60}\)
Is the sequence below part of an arithmetic sequence? In the case that it is part of an arithmetic sequence, find the formula for the \(n\)th term \(a_n\) in the form \(a_n=a_1+d\cdot (n-1)\).
- \(5, 8, 11, 14, 17, \dots\)
- \(-10, -7, -4, -1, 2, \dots\)
- \(-1, 1, -1, 1, -1, 1, \dots\)
- \(18, 164, 310, 474, \dots\)
- \(73.4, 51.7, 30, \dots\)
- \(9, 3, -3, -8, -14, \dots\)
- \(4, 4, 4, 4, 4, \dots\)
- \(-2.72, -2.82, -2.92, -3.02, -3.12, \dots\)
- \(\sqrt{2}, \sqrt{5}, \sqrt{8}, \sqrt{11}, \dots\)
- \(\dfrac{-3}{5}, \dfrac{-1}{10}, \dfrac{2}{5}, \dots\)
- \(a_n=4+5\cdot n\)
- \(a_j=2\cdot j-5\)
- \(a_n=n^2 +8n+15\)
- \(a_k=9\cdot (k+5) +7k-1\)
- Answer
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For the convenience of those who prefer to use \(a_{n}=a+b \cdot n\) as standard form we have provided answers also in that form.
- \(5 + 3(n−1) = 2 + 3n\)
- \(−10 + 3(n−1) = −13 + 3n\)
- no
- no
- \(73.4−21.7(n−1) = 95.1−21.7n\)
- no
- \(4 + 0 ·(n−1) = 4 + 0 ·n\)
- \(-2.72-.1(n-1)=-2.62-.1 n\)
- no
- \(-\dfrac{3}{5}+\dfrac{1}{2}(n-1)=-\dfrac{11}{10}+\dfrac{1}{2} n\)
- \(9 + 5(n − 1) = 4 + 5n\)
- \(−3 + 2(j − 1) = −5 + 2j\)
- no
- \(29 + 16(k − 1) = 13 + 16k\)
Determine the general \(n\)th term \(a_n\) of an arithmetic sequence \(\{a_n\}\) with the data given below.
- \(d=4\), and \(a_{8}=57\)
- \(d=-3\), and \(a_{99}=-70\)
- \(a_1=14\), and \(a_{7}=-16\)
- \(a_1=-80\), and \(a_{5}=224\)
- \(a_{3}=10\), and \(a_{14}=-23\)
- \(a_{20}=2\), and \(a_{60}=32\)
- Answer
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- \(57 + 4(n − 8) = 29 + 4(n − 1) = 25 + 4n\)
- \(−70 − 3(n − 99) = 224 − 3(n − 1) = 227 − 3n\)
- \(14 − 5(n − 1) = 19 − 5n\)
- \(−80 + 76(n − 1) = −156 + 76n\)
- \(10 − 3(n − 3) = 16 − 3(n − 1) = 19 − 3n\)
- \(2+\dfrac{3}{4}(n-20)=-49 / 4+\dfrac{3}{4}(n-1)=-13+\dfrac{3}{4} n\)
Determine the value of the indicated term of the given arithmetic sequence.
- if \(a_1=8\), and \(a_{15}=92\), find \(a_{19}\)
- if \(d=-2\), and \(a_3=31\), find \(a_{81}\)
- if \(a_1=0\), and \(a_{17}=-102\), find \(a_{73}\)
- if \(a_{7}=128\), and \(a_{37}=38\), find \(a_{26}\)
- Answer
-
- \(116\)
- \(187\)
- \(-\dfrac{3621}{8}\)
- \(71\)
Determine the sum of the arithmetic sequence.
- Find the sum \(a_1+\dots +a_{48}\) for the arithmetic sequence \(a_i=4i+7\).
- Find the sum \(\sum_{i=1}^{21}a_i\) for the arithmetic sequence \(a_n=2-5n\).
- Find the sum: \(\sum\limits_{i=1}^{99} (10\cdot i+1)\)
- Find the sum: \(\sum\limits_{n=1}^{200} (-9-n)\)
- Find the sum of the first \(100\) terms of the arithmetic sequence: \(2, 4, 6, 8, 10, 12, \dots\)
- Find the sum of the first \(83\) terms of the arithmetic sequence: \(25, 21, 17, 13, 9, 5, \dots\)
- Find the sum of the first \(75\) terms of the arithmetic sequence: \(2012, 2002, 1992, 1982, \dots\)
- Find the sum of the first \(16\) terms of the arithmetic sequence: \(-11, -6, -1, \dots\)
- Find the sum of the first \(99\) terms of the arithmetic sequence: \(-8, -8.2, -8.4, -8.6, -8.8, -9, -9.2, \dots\)
- Find the sum \(7+8+9+10+\dots+776+777\)
- Find the sum of the first \(40\) terms of the arithmetic sequence: \(5, 5, 5, 5, 5, \dots\)
- Answer
-
- \(5, 040\)
- \(−1, 113\)
- \(49, 599\)
- \(−21, 900\)
- \(10, 100\)
- \(−11, 537\)
- \(123, 150\)
- \(424\)
- \(−1762.2\)
- \(302, 232\)
- \(200\)