Question

Arithmetic Question.

Answer 1

There are two answers

The first answer is 8

The second answer is 15

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Work Shown:

t_1 = first term, t_2 = second term, etc etc

s_n = sum of the first n terms

t_1 = 11/3 ... is given

t_2 = t_1 + d ... add 'd' to the first term to get the second

t_3 = t_2 + d ... repeat last step to second term

t_3 = (t_1+d) + d ... substitution

t_3 = t_1+2d

t_3 = (11/3)+2d ... plug in t_1 = 11/3

t_3 = 3 ... this is given

The third term t_3 is both (11/3)+2d and 3. Equate the two expressions and solve for d.

(11/3)+2d = 3

2d = 3 - (11/3)

2d = (9/3) - (11/3)

2d = (9-11)/3

2d = -2/3

d = (-2/3)*(1/2)

d = -1/3

Turn to the nth term series formula for arithmetic sequences. Plug in the given and found values and isolate n

s_n = (n/2)*(t_1+t_n)

s_n = (n/2)*(t_1+t_1+d(n-1))

s_n = (n/2)*(2*t_1+d(n-1))

20 = (n/2)*(2*(11/3)+(-1/3)*(n-1))

20 = (n/2)*( (22/3)+(-1/3)(n-1))

2*20 =  n*( (22/3)+(-1/3)(n-1) )

40 =  n*( (22/3)+(-1/3)n + (-1/3)(-1) )

40 =  n*( (22/3)+(-1/3)n + (1/3) )

40 =  n*( (23/3)+(-1/3)n )

40 =  (23/3)n+(-1/3)n^2

3*40 =  3*(23/3)n+3*(-1/3)n^2

120 =  23n - n^2

0 =  23n - n^2 - 120

-n^2 + 23n - 120 = 0

n^2 - 23n + 120 = 0

(n - 8)(n - 15) = 0

n-8 = 0 or n-15 = 0

n = 8 or n = 15

If you were to add up the first n = 8 terms, then s_n = 20

If you were to add up the first n = 15 terms, then s_n = 20