Question 1088941
the formulas for arithmetic sereis are:


An = A1 + (n-1) * d


Sn = n * (A1 + An) / 2


you are given:


A7 = 19


S9 = 99


from An = A1 + (n-1) * d, you derive:


A7 = A1 + 6d


since A7 = 19, you derive:


19 = A1 + 6d


from that, you derive:


A1 = 19 - 6d


using the same formula of An = A1 + (n-1) * d, you derive:


A9 = A1 + 8d


you know that A7 = A1 + 6d


if you subtract A7 from A9, you (A1 + 6d) subtract from (A1 + 8d) which gets you:


A9 minus A7 = (A1 + 8d) minus (A1 + 6d) = A1 + 8d - A1 - 6d = 2d


you have:


A9 minus A7 = 2d


solve for A9 to get A9 = A7 + 2d


since you know A7 is equal to 19, then you get:


A9 = 19 + 2d


you now have:


A1 = 19 - 6d


A9 = 19 + 2d


the formula for Sn = n * (A1 + An) / 2


when n = 9, this formula becomes S9 = 9 * (A1 + A9) / 2


you know that S9 = 99 and you derived that A1 = 19 - 6d and A9 = 19 + 2d. 


your formula of S9 = 9 * (A1 + A9) / 2 becomes:


99 = 9 * ((19-6d) + (19+2d)) / 2


simplify this formula to get:


99 = 9 * (38 - 4d) / 2


multiply both sides of this formula by 2 to get:


198 = 9 * (38 - 4d)


simplify further to get:


198 = 342 - 36d


solve for d to get:


d = (342 - 198) / 36


this results in d = 4


now that you know that d = 4, you can solve for A1 and A9.


A1 = 19 - 6d = 19 - 6*4 = 19 - 24 = -5


A9 = 19 + 2*d = 19 + 8 = 27


you have A1 = -5 and A9 = 27


the formula for Sn is Sn = n * (A1 + An) / 2


this becomes S9 = 9 * (-5 + 27) / 2


solve for S9 = 9 * 22 / 2 = 9 * 11 = 99 which is what is should be.


your solution is that the 9th term is 27.


your sequence from first term to 9th term is:


-5 = first term
-1 = second
3 = third
7 = fourth
11 = fifth
15 = sixth
19 = seventh *****
23 = eighth
27 = ninth *****


sum of all 9 terms is 99


everything check out so the solution looks good.


the ninth term is 27.