## anonymous one year ago Find an equation for the nth term of the arithmetic sequence. a19 = -58, a21 = -164

1. anonymous

@Nnesha

2. Nnesha

hey:=) we need common difference and first term to write an equation arithmetic equation $\huge\rm a_n = a_1 + (n-1)d$ where a_1 = first term d=common difference n= term that we have to find

3. Nnesha

given terms are a_{19}= -58 a_{21} = -164 plug this into the equation $\large\rm a_\color{ReD}{{19}}=a_1 +(\color{ReD}{19}-1)d$ 19th term so we substitute n for 19 and a_{19} equal to -58 we can replace a_{19} with -58 $\large\rm -58=a_1 +(\color{ReD}{19}-1)d$ same for a_{21} write an equation when n =21

4. Nnesha

your turn:=) write an equation when n =21 just like i did for n=19 :=)

5. Nnesha

make sense ??

6. anonymous

lets see, -164 = a1 (21 - 1)d ?

7. Nnesha

perfect! $\huge\rm -164 = a_1 +(\color{Red}{21-1})d$ solve parentheses $\large\rm -164 = a_1 +(\color{Red}{20})d$ this is our 2nd equation first one is $\huge\rm -58 = a_1 +(\color{Red}{19-1})d$ $\large\rm -58 = a_1 +(\color{Red}{18})d$

8. Nnesha

$\large\rm -58 = a_1 + \color{Red}{18} d$$\large\rm -164 = a_1 + \color{Red}{20} d$ we will use elimination method to solve this equation :=) so subtract equation one from equation 2 |dw:1444396691116:dw|

9. Nnesha

for elimination method we should change the sign of one of the equation in other words multiply by negative one now we have a_1 - a_1 we can cancel it out |dw:1444396859763:dw| now combine other terms

10. anonymous

2d = -222 ?

11. Nnesha

hmm how did you get 2 d ?

12. Nnesha

remember we multiplied 2nd equation by negative so it's |dw:1444396988124:dw|

13. anonymous

oops, i thought it was 18 okay so, 12d = -106 ?

14. Nnesha

right now solve for d

15. anonymous

-8.833

16. Nnesha

hmm right but keep it in fraction |dw:1444397180870:dw| -53/6 now substitute this into the equation for d to find first term

17. Nnesha

$\huge\rm a_n=a_1 +(n-1)d$ find first term substitute -53/6 for d

18. anonymous

a_n = a_1 + (n - 1)-53/6 ... im kinda lost

19. Nnesha

hmm why ?

20. Nnesha

well n= number of term that we have to find so to find first term substitute n for 1

21. anonymous

a_1 = a_1+ ( n - 1) -53/6

22. Nnesha

i seeeee

23. Nnesha

$\large\rm -58 = a_1 + \color{Red}{18} d$$\large\rm -164 = a_1 + \color{Red}{20} d$ we can use one of these equation to find a_1 not original equation

24. anonymous

Mathway.com is like a caculetor but is a web

25. Nnesha

lets use first one $\rm a_1 +18d =-58$ substitute d for -53/6 to find a_1

26. Nnesha

i'm pretty sure i'm doing something wron ugh sorry let me see

27. anonymous

i see, a_1 + 18(-53/6) = -58 would a_1 be -159 ..?

28. Nnesha

|dw:1444397762260:dw| you were right it's 18

29. anonymous

ooooooooooooh lol!

30. Nnesha

i'm sorry. it was a_1 + (18 -1)d = a_1 + 18d

31. Nnesha

now $\rm a_1 +18(-53) = -58$

32. anonymous

okay sooo, a_1 + 18(-53) = -58 a_1 + -954 = - 58

33. anonymous

a_1 = 896

34. Nnesha

looks good solve for a_1

35. anonymous

soooo then would it be.. a_n = 896 -53 (n-1) ?

36. Nnesha

right :=)

37. anonymous

woohooo! thank you so much!

38. Nnesha

np :=) btw you can distribute (n-1 ) by -53

39. Nnesha

$\rm a_n = 896 -53n+53$ combine like terms