The 10th term of an ap is-27 and the 5th term is -12 what is the 18th find also the sum of its 25th term

2 answers

a+9d = -27
a+4d = -12
subtract:
5d = -15
d = -3

sub into a + 4d = -12 to find a, then use your definitions to find the rest of question.
btw, "..the sum of its 25th term" makes no sense
In AP n-th term is:

an = a + ( n - 1 ) d

where

a = first term

d = common difference

In this case:

a10 = a + 9 d

a10 = - 27

a + 9 d = - 27

a5 = a + 4 d

a5 = - 12

a + 4 d = - 12

Now you must solve system of two equations:

a + 9 d = - 27

a + 4 d = - 12

The solution is:

a = 0 , d = - 3

a18 = a + 17 d

a18 = 0 + 17 ∙ ( - 3 )

a18 = - 51

Sum of first n members of arithmetic progression is:

Sn = n [ 2 a + ( n - 1 ) d ] / 2

S25 = 25 ∙ [ 2 ∙ 0 + 24 ∙ ( - 3 ) ] / 2 = 25 ∙ ( 0 - 72 ) / 2 =

25 ∙ ( - 72 ) / 2 = - 1800 / 2

S25 = - 900