## lavalava 2 years ago The fourth term of an arithmetic sequence is 141, and the seventh term is 132. The first term is _____.

1. satellite73

$a, a+d,a+2d,a+3d,a+4d,...$ you got $$a+3d=141$$ and $$a+6d=132$$

2. satellite73

that means $$132-141=(a+6d)-(a+3d)=3d=-9$$ and so $$d=-3$$

3. lavalava

okay so then d=4?

4. satellite73

actually $$d=-3$$

5. lavalava

6. satellite73

k lets go slow

7. lavalava

8. satellite73

actually before we even start, since $$132<141$$ is it clear that the terms are getting smaller?

9. satellite73

in other words, "$$d$$" the "common difference" must be negative right?

10. lavalava

okay

11. satellite73

if we call the first term $$a$$ then the second term is $$a+d$$ for some $$d$$ and the third term is $$a+2d$$, the fourth term is $$a+3d$$ the fifth term is $$a+4d$$ etc

12. satellite73

in other words, you keep adding $$d$$ to each term to get the next term, with the sophistication that you might be "adding" a negative number

13. lavalava

okay

14. satellite73

The fourth term of an arithmetic sequence is 141 tells you that $$a+3d=141$$

15. lavalava

okay

16. satellite73

you see that it is the fourth term, so it is $$a+3d$$ not $$a+4d$$

17. lavalava

ohhh

18. satellite73

and the seventh term is 132 means $a+6d=132$

19. satellite73

from these two pieced of information we can solve for $$d$$ and then solve for $$a$$

20. satellite73

*pieces

21. lavalava

okay

22. satellite73

a bit of algebra shows that $a+6d-(a+3d)=3d$ right?

23. lavalava

ohh okay i get it..

24. satellite73

so we see that $3d=132-141=-9$

25. lavalava

mhm

26. satellite73

so far so good?

27. lavalava

and so since it is 7-4= 3 then -9/3 would be -3 right? giving us the difference

28. satellite73

yeah $$-3$$ is the difference

29. satellite73

what you said

30. lavalava

ohhh okay!!! i get it!! :D

31. satellite73

you are still not done though right?

32. satellite73

33. lavalava

hmm well pluggin in the difference and then your equation... a4=141 a3=141+3=144 a2=144+3=147 a1=147+3=150 so then the first term would be 150 right? i just did it backwards

34. satellite73

yeah i guess so i would have said $$a+3\times (-3)=141$$or $a-9=141$ making $$a=150$$ your method means you understand what is going on, which is good but you certainly wouldn't want to use that if you had say $$a_{75}$$ and wanted $$a_1$$

35. lavalava

:D yay thank you!!! :D ohh okay... ill keep in mind that equation!! thank you soo much!

36. satellite73

yw