The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers.
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Step-by-Step Solution
Step 1: Represent the numbers in A.P.
Let the three numbers in an Arithmetic Progression (A.P.) be represented as a - d, a, and a + d. Here, a is the middle term and d is the common difference.
Step 2: Formulate equations from given conditions
We are given two conditions: first, the product of the three numbers is 224. This gives us the equation (a−d)a(a+d)=224. Second, the largest number (a + d) is 7 times the smallest number (a - d), which gives us a+d=7(a−d).
Step 3: Solve the second equation for 'a' in terms of 'd'
Let's simplify the second equation: a+d=7(a−d). Distribute the 7 on the right side to get a+d=7a−7d. Rearranging the terms, we get 8d=6a, which simplifies to a=34d.
Step 4: Substitute 'a' into the first equation and solve for 'd'
Now, substitute a=34d into the product equation (a−d)a(a+d)=224. This simplifies to (31d)(34d)(37d)=224. Multiplying these terms gives 2728d3=224. Solving for d3, we get d3=216. Taking the cube root, we find d=6.
Step 5: Find the value of 'a'
With d=6, we can find the value of a using the relation a=34d. Substituting d=6, we get a=34(6)=8.
Step 6: Determine the three numbers
Finally, substitute the values of a=8 and d=6 back into our expressions for the three numbers: a - d, a, and a + d. This gives us 8−6=2, 8, and 8+6=14. So the three numbers are 2, 8, and 14.