Answers:
[tex]t_{10} = -22 \ \text{ and } S_{10} = -85[/tex]
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Explanation:
[tex]t_1 = \text{first term} = 5\\t_2 = \text{first term}-3 = t_1 - 3 = 5-3 = 2[/tex]
Note we subtract 3 off the previous term (t1) to get the next term (t2). Each new successive term is found this way
[tex]t_3 = t_2 - 3 = 2-3 = -1\\t_4 = t_3 - 3 = -1-3 = -4[/tex]
and so on. This process may take a while to reach [tex]t_{10}[/tex]
There's a shortcut. The nth term of any arithmetic sequence is
[tex]t_n = t_1+d(n-1)[/tex]
We plug in [tex]t_1 = 5 \text{ and } d = -3[/tex] and simplify
[tex]t_n = t_1+d(n-1)\\t_n = 5+(-3)(n-1)\\t_n = 5-3n+3\\t_n = -3n+8[/tex]
Then we can plug in various positive whole numbers for n to find the corresponding [tex]t_n[/tex] value. For example, plug in n = 2
[tex]t_n = -3n+8\\t_2 = -3*2+8\\t_2 = -6+8\\t_2 = 2[/tex]
which matches with the second term we found earlier. And,
[tex]tn = -3n+8\\t_{10} = -3*10+8\\t_{10} = -30+8\\t_{10} = \boldsymbol{-22} \ \textbf{ is the tenth term}[/tex]
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The notation [tex]S_{10}[/tex] refers to the sum of the first ten terms [tex]t_1, t_2, \ldots, t_9, t_{10}[/tex]
We could use either the long way or the shortcut above to find all [tex]t_1[/tex] through [tex]t_{10}[/tex]. Then add those values up. Or we can take this shortcut below.
[tex]Sn = \text{sum of the first n terms of an arithmetic sequence}\\S_n = (n/2)*(t_1+t_n)\\S_{10} = (10/2)*(t_1+t_{10})\\S_{10} = (10/2)*(5-22)\\S_{10} = 5*(-17)\\\boldsymbol{S_{10} = -85}[/tex]
The sum of the first ten terms is -85
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As a check for [tex]S_{10}[/tex], here are the first ten terms:
- t1 = 5
- t2 = 2
- t3 = -1
- t4 = -4
- t5 = -7
- t6 = -10
- t7 = -13
- t8 = -16
- t9 = -19
- t10 = -22
Then adding said terms gets us...
5 + 2 + (-1) + (-4) + (-7) + (-10) + (-13) + (-16) + (-19) + (-22) = -85
This confirms that [tex]S_{10} = -85[/tex] is correct.
