Question

the third term of an arithmetic sequence is 14 and the ninth term is -1. Find the first four terms of the sequence

Answer 1

ninth -1, third 14
it goes now by 15 in six steps; step 15/6 or 2.5
third through tenth terms
14, 11.5, 9, 6.5, 4, 1.5, -1, -3.5,...

check that

Answer 2

third term is 14 and ninth term is -1

14 - 2.5 = 11.5 > 4th term
11.5 - 2.5 = 9 > 5th term
9 - 2.5 = 6.5 > 6th term
6.5 - 2.5 = 4 > 7th term
4 - 2.5 = 1.5 > 8th term
1.5 - 2.5 = -1 > 9th term

The terms are decreasing by 2.5 > (-2.5)
To go from one term to the next, subtract 2.5.

The common difference is -2.5
d = -2.5

Arithmetic Sequence Formula:
Tn = Tn + d(n-1)

a = 1st term
n = nth term
d = common difference

We don't know the first term yet!

Tn = T1 + d(n-1)

Substitute 3 for n, and -2.5 for d

T3 = T1 + -2.5(3-1)

Now substitute 14 for T3

14 = T1 + -2.5(2)
14 = T1 -5
14 + 5 = T1 -5 +5
19 = T1

T1 = 19
T2 = 16.5
T3 = 14
T4 = 11.5
Now to find a term:

Tn = T1 + d(n-1)

T2 = 19 + -2.5(2-1) T3 = 19 + -2.5(3-1)
T2 = 19 + -2.5(1) T3 = 19 + -2.5(2)
T2 = 19 + -2.5 T3 = 19 + -5
T2 = 16.5 T3 = 14
__________________________

You can also use this formula:

Tn = dn + 21.5

T1 = -2.5(1) + 21.5
T1 = 19

T4 = -2.5(4) + 21.5
T4 = 11.5

T9 = -2.5(9) + 21.5
T9 = -1