SOLUTION: Thank you ikleyn for solving my question about the theater problem. The sum of the the first nine terms of an arithmetic sequence is 99, and its seventh term is 19. What is its

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Question 1088941: Thank you ikleyn for solving my question about the theater problem.
The sum of the the first nine terms of an arithmetic sequence is 99, and its seventh term is 19. What is its ninth term?
So far my solution is(Im unsure of my solution)
a7=19(Seventh term);a9=99(First nine terms); a9=?(Ninth term)
Sn=n/2(a1+an)
Sn=9/2(a1+a9)
99=4.5(a1 + ______) [Not sure what to add here]
a9=a6+a7+6d
a9=a7+19+6d
a9=a1+7d+19+6d
a9=a1+19+13d
2a1+19+13d
Basically I am not sure of my formula. If you don't seem to understand the problem, I have a sample problem(with solution) identical to this problem you can request it if you want.
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
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.









Answer by MathTherapy(10556)   (Show Source): You can put this solution on YOUR website!
Thank you ikleyn for solving my question about the theater problem.
The sum of the the first nine terms of an arithmetic sequence is 99, and its seventh term is 19. What is its ninth term?
So far my solution is(Im unsure of my solution)
a7=19(Seventh term);a9=99(First nine terms); a9=?(Ninth term)
Sn=n/2(a1+an)
Sn=9/2(a1+a9)
99=4.5(a1 + ______) [Not sure what to add here]
a9=a6+a7+6d
a9=a7+19+6d
a9=a1+7d+19+6d
a9=a1+19+13d
2a1+19+13d
Basically I am not sure of my formula. If you don't seem to understand the problem, I have a sample problem(with solution) identical to this problem you can request it if you want.
You were given the sum of the 1st 9 terms of the AP, and the 7th term so that the 1st term and d: the common difference, can be determined. 
Therefore, I would NOT USE the SUM of an AP formula you used but this instead: 
 ---- Substituting 9 for n
 --------- Substituting 
 ---------- Cross-multiplying

 ======>  -------- eq (i)


 -------- Substituting 7 for n
 ------------- Substituting  ------- eq (ii)
8 = 2d ------ Subtracting eq (i) from eq (ii)


 --------- Substituting 4 for d in eq (i)



 ------- Substituting – 5 for a1, and 4 for d

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