The 3th and 9th term of a G.p are 54and 39.3660 respectively find
a. 6th term
b. Sum of the 4th and 7th
Let the first term of the geometric progression be "a" and the common ratio be "r".
We are given that the 3rd term is 54, so we can write the equation:
a*r^2 = 54
We are also given that the 9th term is 39.3660, so we can write the equation:
a*r^8 = 39.3660
Now, we can divide these two equations to eliminate "a":
(a*r^2) / (a*r^8) = 54 / 39.3660
Simplifying, we get:
r^6 = (54 / 39.3660)
Taking the sixth root of both sides, we get:
r = (54 / 39.3660)^(1/6)
Using a calculator, we find that r ≈ 1.1497.
Now, we can substitute this value of r into the first equation to solve for "a":
a*(1.1497)^2 = 54
Simplifying, we get:
a = 54 / (1.1497)^2
Using a calculator, we find that a ≈ 41.1979.
a. To find the 6th term, we can use the formula for the nth term of a geometric progression:
6th term = a * r^(6-1)
Substituting our values, we get:
6th term = 41.1979 * (1.1497)^(6-1)
Using a calculator, we find that the 6th term ≈ 100.7873.
b. To find the sum of the 4th and 7th terms, we can use the formula for the sum of the first n terms of a geometric progression:
Sum = a * (1 - r^n) / (1 - r)
Sum of 4th and 7th terms = a * (1 - r^3) / (1 - r) + a * (1 - r^6) / (1 - r)
Substituting our values, we get:
Sum of 4th and 7th terms ≈ 41.1979 * (1 - (1.1497)^3) / (1 - 1.1497) + 41.1979 * (1 - (1.1497)^6) / (1 - 1.1497)
Using a calculator, we find that the sum of the 4th and 7th terms ≈ 174.6782.