The 5th term of an arithmetic series is -18 and the 9th term is 4. Find the sum of the first 20 terms.

Question

campbell_st · Answer

do you know the formula for a term in an arithmetic sequence..?

anonymous · Answer

tn=ti+(n-1)d

campbell_st · Answer

ok... I use a instead of ti.... for the 1st value

so you know

-18 = a +  (5 - 1) d      or      -18 = a + 4d

4 = a + ( 9 -1) d             or           4 = a + 8d

you need to solve the equations 

-18 = a + 4d
   4 = a + 8d 

to find the 1st term and the common difference... 

then you will be able to find the sum of the 1st 20 terms

anonymous · Answer

Yup, I did that- and I got -3.5 for the d value, then I plugged that into on of the equations and got 32 for the a value (are these values correct?) and then I'm unsure as to where to go from there...

campbell_st · Answer

so using elimination you may want to check my calculations...

      -18 = a + 4d
-        4 = a + 8d
----------------
      -22 = -4d

d = -5.5

campbell_st · Answer

oops d = 5.5

campbell_st · Answer

once you know the 1st term and the common difference the sum of an arithmetic series is

$$S_{n} = \frac{n}{2}[2a + (n - 1)d]$$

just substitute and evaluate

anonymous · Answer

Ah yes indeed, thank you for that correction! 
Alright, but how would we substitute into the formula, given the fact that we have two terms?

anonymous · Answer

Hello?

campbell_st · Answer

ok so a = -40 and d = 5.5 

so 

$$S_{20} = \frac{20}{2}[2 \times -40 + (20 - 1) \times 5.5]$$

now just calculate it out.

anonymous · Answer

Cool, may I ask what formula that is though? Or did you just derive it for this problem in particular?

Because I thought the following equation is the one you use for arithmetic$$Sn=(n/2)(a+tn)$$ series:

campbell_st · Answer

well its the same formula as you use

you use n/2( 1st + last) 

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