Find the G.p. in which 4th terms is 3 and the 7th term is 8/9.
T7/T4 = r^3 = (8/9)/3 = 8/27
so, r = 2/3
Now you can find the 1st term and thus whatever terms you want.
To find the geometric progression (G.P.), we need to determine the common ratio (r) first. We are given that the 4th term is 3 and the 7th term is 8/9.
The formula for the nth term of a G.P. is given by:
an = a1 * r^(n-1)
Let's label the 4th term as a4 and the 7th term as a7. Using the formula above, we can set up the following equations:
a4 = a1 * r^(4-1) = a1 * r^3 (equation 1)
a7 = a1 * r^(7-1) = a1 * r^6 (equation 2)
We are given that a4 = 3 and a7 = 8/9. Let's substitute the values into the equations:
3 = a1 * r^3 (equation 1)
8/9 = a1 * r^6 (equation 2)
To solve for r, we can divide equation 2 by equation 1:
(8/9) / 3 = (a1 * r^6) / (a1 * r^3)
8/27 = r^3 / r^6
Simplifying further:
8/27 = 1/r^3
Now, we can find r by taking the cube root of both sides:
∛(8/27) = r
Simplifying the cube root:
2/3 = r
So, the common ratio (r) is 2/3.
Now that we have the common ratio, we can find the first term (a1) using equation 1. Using the value of r as 2/3 and a4 as 3:
3 = a1 * (2/3)^3
3 = a1 * (8/27)
Multiplying both sides by (27/8):
(3 * 27) / 8 = a1
81/8 = a1
Therefore, the first term (a1) of the geometric progression is 81/8 and the common ratio (r) is 2/3.