A GP has a first term of a, a common ratio of [tex]$r$[/tex] and its 6th term is 768. Another GP has a first term of a, a common ratio of [tex]$6r$[/tex] and its 3rd term is 3456. Evaluate a and [tex]$r$[/tex].
Answer (2)
Establish two equations based on the given information about the two geometric progressions: a r 5 = 768 and 36 a r 2 = 3456 .
Divide the second equation by the first to eliminate 'a' and solve for 'r': r 3 36 = 2 9 , which simplifies to r 3 = 8 .
Calculate r by taking the cube root: r = 2 .
Substitute the value of r back into the first equation to solve for 'a': a ( 2 ) 5 = 768 , which gives a = 24 .
The final answer is: a = 24 , r = 2 .
Explanation
Problem Analysis We are given two geometric progressions (GPs). Our goal is to find the values of 'a' (the first term) and 'r' (the common ratio) for these GPs. Let's analyze the given information.
First GP Equation For the first GP, we know that the 6th term is 768. The formula for the nth term of a GP is given by T n = a n − 1 , where T n is the nth term, a is the first term, and r is the common ratio. Therefore, for the first GP, we have: T 6 = a 6 − 1 = a 5 = 768
Second GP Equation For the second GP, the first term is 'a', the common ratio is '6r', and the 3rd term is 3456. Using the same formula, we have: T 3 = a ( 6 r ) 3 − 1 = a ( 6 r ) 2 = 36 a 2 = 3456
Solving for r Now we have two equations: Equation (1): a 5 = 768 Equation (2): 36 a 2 = 3456 To solve for 'a' and 'r', we can divide equation (2) by equation (1): a 5 36 a 2 = 768 3456 Simplifying the equation, we get: 3 36 = 768 3456 = 4.5 = 2 9
Calculating r Now, we solve for r 3 :
3 = 2 9 36 = 9 36 im es 2 = 8 Taking the cube root of both sides, we get: r = 3 8 = 2
Calculating a Now that we have the value of r, we can substitute it back into equation (1) to find the value of 'a': a ( 2 ) 5 = 768 32 a = 768 a = 32 768 = 24
Final Answer Therefore, the values of 'a' and 'r' are: a = 24 r = 2
Examples
Geometric progressions are useful in many areas of mathematics and in real-world applications. For example, understanding GPs can help calculate compound interest on investments, model population growth, or analyze the decay of radioactive substances. In finance, if you invest a certain amount of money each year with a fixed interest rate, the total value of your investment over time can be calculated using the formula for the sum of a GP. Similarly, in biology, the growth of a bacterial colony under ideal conditions can be modeled as a GP, where each generation increases by a constant factor.
The values of 'a' and 'r' in the geometric progressions are a = 24 and r = 2. This was determined by setting up equations based on the terms of the GPs and solving for the unknowns. Both results were verified by substituting back into the equations.
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