Three numbers form a GP. If the first and third numbers are 5 and 245 respectively, find two possible values for the middle number.
Answer (2)
Represent the geometric progression as a , a r , a r 2 .
Substitute the given values a = 5 and a r 2 = 245 .
Solve for the common ratio r , obtaining r = ± 7 .
Calculate the two possible middle terms a r , which are 35 and − 35 . The final answer is 35 , − 35 .
Explanation
Understanding the Problem We are given that three numbers form a geometric progression (GP). The first number is 5 and the third number is 245. Our goal is to find the two possible values for the middle number.
Setting up the Equations Let the three numbers in the geometric progression be a , a r , a r 2 , where a is the first term and r is the common ratio. We are given that the first term a = 5 and the third term a r 2 = 245 .
Solving for r^2 We can substitute the value of a into the equation for the third term: 5 r 2 = 245 Now, we solve for r 2 by dividing both sides by 5: r 2 = 5 245 = 49
Finding the Common Ratio To find the possible values of r , we take the square root of both sides: r = ± 49 = ± 7 So, the common ratio r can be either 7 or -7.
Calculating the Middle Numbers The middle number in the geometric progression is a r . We have two possible values for r , so we calculate the middle number for each case:
If r = 7 , the middle number is 5 × 7 = 35 .
If r = − 7 , the middle number is 5 × ( − 7 ) = − 35 .
Final Answer Therefore, the two possible values for the middle number are 35 and -35.
Examples
Geometric progressions are useful in many real-world scenarios, such as calculating compound interest, modeling population growth or decay, and analyzing the behavior of certain physical systems. For example, if you invest $5000 in an account that earns 7% interest compounded annually, the amounts you have each year form a geometric progression. Understanding geometric progressions helps in predicting future values and making informed decisions in finance and other fields.
The middle number of the geometric progression with the first term 5 and the third term 245 can be either 35 or -35, depending on the common ratio chosen. This is calculated by finding the square root of the ratio of the third term to the first term. Thus, the possible middle numbers are 35 and -35.
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