Pre Calculus 11 - Arithmetic Series

Question

What is the first term of a 10-term arithmetic series with t2=10 and S=120? Leave your answer in exact form.

Sadly, I don't even know where to start with this question. I understand that there is probably a rather complicated equation that goes along with answering this question. Any help with getting the proper equation/answer would be greatly appreciated.

Thank you

Answer 1

The unknown common difference is d

1st term : x

2nd term 10

3rd term: x + 2d

4th term: x + 3d

...

10th term: x + 9d

X + 10 + X+ 2d + X+3d + x+4d + x + 5d + x + 6d + x +7d + x + 8d + x + 9d = 120

9x + 44d + 10 = 120

9x + 44d = 110 <--- please label this equation ALPHA; we need it soon!!!

THe second term is 10 and it differs from the 1st term by d:

x+d = 10 ---> d = 10 - x <--- please label this equation BETA; we need it soon !!!

9x + 44(10 - x) = 110 <--- substitutes into equation ALPHA label and in bold above

9x + 440 - 44x = 110

-35x = -330

x = 9 and 3/7 = 66/7

Per equation BETA in bold above,

the common difference is then d = 10 - x = 10 - 66/7 = (70-66)/7 = 4/7

1st term: 9 and 3/7

2nd term: 10

3rd term : 10 and 4/7

4th term : 11 and 1/7

5th term: 11 and 5/7

6th term: 12 and 2/7

7th term: 12 and 6/7

8th term: 13 and 3/7

9th term: 14

10th term: 14 and 4/7

total: 116 and 28/7 = 116+4 = 120

So the first term is 9 and 3/7 = 66/7 and the common difference is 4/7

Tn = 66/7 + (4/7)(n-1), n=1,2,3,....

Answer 2

S_n = (n / 2)(2a₁ + (n - 1)d)

a₁ = a₂ - d

a₁ = 10 - d

n = 10

120 = 5(20 - 2d + 9d)

24 = 20 - 10d + 18d

4 = 8d

0.5 = d

a₁ = 9.5