Question

Dont really understand how to do this

Answer 1

Answers:

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

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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}`

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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

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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.