The third term of geometric progression(GP)is 24 and it’s seventh term is 4 20/27. Find:
(i) the first term (ii) the common ratio
(i) Let the first term be a and the common ratio be r.
Given that the third term is 24, we have:
a*r^2 = 24 (1)
Given that the seventh term is 4 20/27, we have:
a*r^6 = 4 20/27 (2)
Expressing 4 20/27 as an improper fraction, we get:
4 20/27 = 4 + 20/27 = 108/27 + 20/27 = 128/27
Now, substitute (1) and (2) into the equations above and solve for a and r:
From equation (1):
a*r^2 = 24
a = 24/r^2 (3)
From equation (2):
a*r^6 = 128/27
(24/r^2)*r^6 = 128/27
24r^4 = 128/27
r^4 = (128/27)/24
r^4 = (128/27) * (1/24)
r = (128/648)^(1/4)
r = 2/3
Substitute r = 2/3 into equation (3):
a = 24/(2/3)^2
a = 24 / (4/9)
a = 24 * 9/4
a = 54
Therefore, the first term is 54 and the common ratio is 2/3.