Question 1209845
Let's break down this problem step-by-step.

Understanding Geometric Progressions

A geometric progression (GP) is a sequence where each term is found by multiplying the previous term by a constant called the common ratio (r). The general form of a GP is:

a, ar, ar^2, ar^3, ...

where:

a is the first term
r is the common ratio
Given Information

The second term (ar) is 18.
The seventh term (ar^6) is 4374.
Finding a and r

Set up equations:

ar = 18 (Equation 1)
ar^6 = 4374 (Equation 2)
Divide Equation 2 by Equation 1:

(ar^6) / (ar) = 4374 / 18
r^5 = 243
Solve for r:

r = ∛⁵243 = 3
Substitute r back into Equation 1 to find a:

a(3) = 18
a = 18 / 3 = 6
Finding the Fourth and Eighth Terms

Fourth term (ar^3):

ar^3 = 6 * 3^3 = 6 * 27 = 162
Eighth term (ar^7):

ar^7 = 6 * 3^7 = 6 * 2187 = 13122
Finding the Sum

We are asked to find the sum of the fourth and eighth terms when the difference is 3. We have already determined that r=3, so we can use the values calculated above.

Sum = 162 + 13122 = 13284
Answer

The sum of the fourth and eighth term is 13284.