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Find the largest of three numbers in Arithmetic Progression whose sum is 87 and whose product is 24273
The largest of the three numbers in the arithmetic progression is 31.
Let’s denote the three numbers in the arithmetic progression as a-d, a, and a+d, where a is the middle term and d is the common difference.
Given that the sum of these three numbers is 87, we have the equation:
(a – d) + a + (a + d) = 87
3a = 87
a = 29
Now, since the product of these three numbers is 24273, we have the equation:
(a – d) * a * (a + d) = 24273
(29 – d) * 29 * (29 + d) = 24273
Now, we can solve this equation to find the value of d:
(29 – d) * 29 * (29 + d) = 24273
(29^2 – d^2) * 29 = 24273
(841 – d^2) * 29 = 24273
841 – d^2 = 24273/29
841 – d^2 = 837
-d^2 = -4
d^2 = 4
d = 2
Now that we have found a and d, we can find the largest number, which is a + d: a + d = 29 + 2 = 31
So, the largest of the three numbers in the arithmetic progression is 31.
What is Arithmetic Progression?
Arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). The general form of an arithmetic progression is given by:
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a, a + d, a + 2d, a + 3d, …
Where:
- a is the first term of the sequence,
- d is the common difference between consecutive terms,
- a + nd represents the nth term of the sequence.
Arithmetic progressions are widely used in various mathematical contexts, including algebra, number theory, and calculus. They play a fundamental role in understanding series, sequences, and solving mathematical problems in diverse fields.
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