find the 15th term of a linear sequence, whose 5th term is 30 and 20th term is 75.

Answer 1

T15 is 2/3 of the way from T5 to T20
So, its value will be 2/3 of the way from 30 to 75.

Or, note that
T20 = T5 + 15d
So, 15d = 75-30 = 45
d = 15

T15 = T5 + 10d

Answer 2

Recall the formular
Tn=a+(n-1)d
n=15
T15=a+(15-1)d
=a+14d
but 5th term=30
30=a+(5-1)d
30=a+4d ........(i)
20th term=75
75=a+(20-1)d
75=a+19d........(ii)
from (i) make a the subject of the formular
30=a+4d
a=30-4d.........(iii)
sub (iii) into (ii)
75=30-4d+19d
C.L.T
75-30=-4d+19d
45/15=15d/15
d=3
sub d into (i)
30=a+4*3
30=a+12
a=30-12
a=18
therefore T15=18+14*3
=60