In a space exploration article, a journalist describes a sequence of five real numbers representing the distance of planets from their star, forming a geometric progression. If the first term is 4 million km and the common ratio is 1.5, solve for the fifth term of this sequence.
["Exploring Cosmic Distances: The Geometric Progression of Planets in a Distant Solar System", "In the vast realm of space exploration, planetary systems often reveal fascinating mathematical patterns—one such pattern appears in the distances of planets from their host star. Recent astronomical modeling suggests that a newly observed exoplanetary system exhibits a striking sequence: the distances of five planets from their central star follow a geometric progression, a concept that blends astronomy with elegant mathematical structure.", "Let’s dive into the numbers behind this cosmic rhythm.", "A geometric progression is a sequence of values where each term after the first is found by multiplying the previous one by a constant called the common ratio. According to recent exploratory data, the distance of the first planet from the star is 4 million kilometers. With a common ratio of 1.5, the sequence unfolds as follows:", "- First term ((a_1)) = 4 million km\n- Common ratio ((r)) = 1.5", "Using the formula for the (n)-th term of a geometric progression:\n[\na_n = a_1 \ imes r^{(n-1)}\n]\nwe solve for the fifth term ((a_5)):\n[\na_5 = 4 \ imes (1.5)^{4}\n]\nCalculating (1.5^4):\n[\n1.5^2 = 2.25 \quad \ ext{and} \quad 1.5^4 = (1.5^2)^2 = 2.25^2 = 5.0625\n]\nThus,\n[\na_5 = 4 \ imes 5.0625 = 20.25\n]", "So, the fifth planet lies at a distance of 20.25 million kilometers from its star—a clear and measurable cosmic milestone shaped by a consistent geometric progression.", "This pattern mirrors the natural order found in space and underscores how mathematics helps uncover the hidden laws governing planetary systems. As telescopes and simulations advance, identifying such sequences not only enhances our understanding of alien worlds but also connects Earth-bound science with the grandeur of the universe.", "Stay tuned for more discoveries where astronomy meets precision. Whether probing the edges of our solar neighborhood or scanning distant star systems, the language of numbers continues to guide our journey beyond the stars."]
