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Dont really understand how to do this

Dont really understand how to do this-example-1
User Gosom
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Answers:


t_(10) = -22 \ \text{ and } S_(10) = -85

========================================================

Step-by-step explanation:


t_1 = \text{first term} = 5\\t_2 = \text{first term}-3 = t_1 - 3 = 5-3 = 2

Note we subtract 3 off the previous term (t1) to get the next term (t2). Each new successive term is found this way


t_3 = t_2 - 3 = 2-3 = -1\\t_4 = t_3 - 3 = -1-3 = -4

and so on. This process may take a while to reach
t_(10)

There's a shortcut. The nth term of any arithmetic sequence is


t_n = t_1+d(n-1)

We plug in
t_1 = 5 \text{ and } d = -3 and simplify


t_n = t_1+d(n-1)\\t_n = 5+(-3)(n-1)\\t_n = 5-3n+3\\t_n = -3n+8

Then we can plug in various positive whole numbers for n to find the corresponding
t_n value. For example, plug in n = 2


t_n = -3n+8\\t_2 = -3*2+8\\t_2 = -6+8\\t_2 = 2

which matches with the second term we found earlier. And,


tn = -3n+8\\t_(10) = -3*10+8\\t_(10) = -30+8\\t_(10) = \boldsymbol{-22} \ \textbf{ is the tenth term}

---------------------

The notation
S_(10) refers to the sum of the first ten terms
t_1, t_2, \ldots, t_9, t_(10)

We could use either the long way or the shortcut above to find all
t_1 through
t_(10). Then add those values up. Or we can take this shortcut below.


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}

The sum of the first ten terms is -85

-----------------------

As a check for
S_(10), 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
S_(10) = -85 is correct.

User Needpoule
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