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Question 1209845: The second and seventh term of a geometric progression are 18 and 4374 respectively. Find the sum of the fourth and the eighth term when the difference is 3
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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