Question

The 5th and 10th terms of a linear sequence are -12 and -27 respectively.Find first term and the common difference of the sequence

Answer 1

use your definitions:
a+4d = -12
a+9d = -27
subtract them,
5d = -15
d = -3
back into a+4d = -12
a - 12 = -12
a = 0

check: sequence is 0,-3,-6,-9,-12,-15,-18,-21,-24,-27,-30
yup, we got it

Answer 2

Linear sequence is other name for arithmetic sequence.

For arithmetic sequence nth term:

an = a1 + ( n - 1 ) d

where:

a1 = initial term of arithmetic progression

d = common difference

In this case:

a5 = a1 + ( 5 - 1 ) d

a5 = a1 + 4 d

a10 = a1 + ( 10 - 1 ) d

a10 = a1 + 9 d

so:

a5 = -12

- 12 = a1 + 4 d

a10 = - 27

- 27 = a1 + 9 d

Now you must solve system:

- 12 = a1 + 4 d

- 27 = a1 + 9 d

Try it.

The solutions are:

a1 = 0

d = - 3

Your sequence:

0 , - 3 , - 6 , - 9 , - 12 , -15 , - 18 , - 21 , - 24 , - 27...