The sum of three numbers in an A.P. is 15. If 1, 4, and 19 are added to them respectively, the resulting numbers are in G.P. Find the initial three numbers.
Answer (2)
Define the three numbers in A.P. as a − d , a , and a + d , and use the given sum to find a = 5 .
Add 1, 4, and 19 to the numbers, resulting in 6 − d , 9, and 24 + d , which are in G.P.
Set up the proportion 6 − d 9 = 9 24 + d and solve the quadratic equation d 2 + 18 d − 63 = 0 , obtaining d = 3 or d = − 21 .
Find the two possible sets of numbers: 2, 5, 8 and 26, 5, -16. The final answer is 2 , 5 , 8 or 26 , 5 , − 16 .
Explanation
Understanding the Problem We are given that the sum of three numbers in arithmetic progression (A.P.) is 15. Also, if we add 1, 4, and 19 to these numbers respectively, the resulting numbers are in geometric progression (G.P.). Our goal is to find the initial three numbers in A.P.
Finding the Middle Term Let the three numbers in A.P. be a − d , a , and a + d , where a is the middle term and d is the common difference. Since their sum is 15, we have: ( a − d ) + a + ( a + d ) = 15 This simplifies to: 3 a = 15 a = 5 So, the three numbers are 5 − d , 5 , and 5 + d .
Setting up the Geometric Progression When we add 1, 4, and 19 to these numbers respectively, we get the numbers 5 − d + 1 = 6 − d , 5 + 4 = 9 , and 5 + d + 19 = 24 + d . Since these resulting numbers are in G.P., the ratio between consecutive terms must be constant. Therefore: 6 − d 9 = 9 24 + d
Forming a Quadratic Equation Cross-multiplying, we get: 81 = ( 6 − d ) ( 24 + d ) Expanding the right side gives: 81 = 144 + 6 d − 24 d − d 2 81 = 144 − 18 d − d 2 Rearranging the terms, we obtain a quadratic equation in d :
d 2 + 18 d − 63 = 0
Solving for the Common Difference We can solve this quadratic equation for d using the quadratic formula or by factoring. Let's factor the quadratic: ( d + 21 ) ( d − 3 ) = 0 This gives us two possible values for d : d = 3 or d = − 21 .
Finding the Numbers If d = 3 , the three numbers in A.P. are: 5 − 3 = 2 , 5 , 5 + 3 = 8 The numbers are 2, 5, and 8. Adding 1, 4, and 19 to them gives 3, 9, and 27, which are in G.P. since 9/3 = 27/9 = 3 .
If d = − 21 , the three numbers in A.P. are: 5 − ( − 21 ) = 26 , 5 , 5 + ( − 21 ) = − 16 The numbers are 26, 5, and -16. Adding 1, 4, and 19 to them gives 27, 9, and 3, which are in G.P. since 9/27 = 3/9 = 1/3 .
Final Answer Thus, the initial three numbers in A.P. can be either 2, 5, and 8 or 26, 5, and -16.
Examples
Arithmetic and geometric progressions are not just abstract mathematical concepts; they appear in various real-world scenarios. For instance, the seats in a theater often follow an arithmetic progression, where each row has a fixed number of seats more than the previous row. Geometric progressions can model compound interest, where an initial investment grows by a fixed percentage over time. Understanding these progressions helps in predicting patterns and making informed decisions in finance, design, and other fields.
The three numbers in A.P. that sum to 15 can be either (2, 5, 8) or (26, 5, -16). In both cases, adding 1, 4, and 19 results in numbers that are in geometric progression. Thus, these represent valid solutions to the problem.
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