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Consider the expression
(a + 2b) (b + 3c) (c + 4d) .........(y + 26z) (z + a) / abc...z
Where a, b, c, d , ...... z are positive integers. The minimum value of the expression is
(a + 2b) (b + 3c) (c + 4d) .........(y + 26z) (z + a) / abc...z
Where a, b, c, d , ...... z are positive integers. The minimum value of the expression is
- a)226 √27! - 1
- b)226 √26!
- c)227 √26!
- d)27!
- e)227√27!
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the expression(a + 2b) (b + 3c) (c + 4d) .........(y + 26z) (...
By AM-GM inequality
a + 2b / 2 ≥ √2ab
∴ a + 2b / √ab ≥ 2√2
Similarly,
b + 3c / √bc ≥ 2√3
c + 4d / √cd ≥ 2√4
y + 26z / √yz ≥ 2√26
z + a / √za ≥ 2
Multiplying, we get
(a + 2b) (b + 3c) (c + 4d)......(y + 26z) (z +a) / abc...z ≥ 226 x √26!
Hence, option 2.
a + 2b / 2 ≥ √2ab
∴ a + 2b / √ab ≥ 2√2
Similarly,
b + 3c / √bc ≥ 2√3
c + 4d / √cd ≥ 2√4
y + 26z / √yz ≥ 2√26
z + a / √za ≥ 2
Multiplying, we get
(a + 2b) (b + 3c) (c + 4d)......(y + 26z) (z +a) / abc...z ≥ 226 x √26!
Hence, option 2.
Most Upvoted Answer
Consider the expression(a + 2b) (b + 3c) (c + 4d) .........(y + 26z) (...
Let's break down the expression step by step.
First, we have (a 2b). This means that the value of the expression is a multiplied by 2b.
Next, we have (b 3c). This means that the value of the expression is now a multiplied by 2b, and then multiplied by 3c.
Continuing this pattern, we have (c 4d), (d 5e), and so on, until we reach (y 26z). Finally, we have (z a).
To find the minimum value of the expression, we want to minimize each term in parentheses. Since a, b, c, d, ..., z are positive integers, the minimum value for each term is when the smaller number is 1 and the larger number is the next consecutive integer.
So, the minimum value for (a 2b) is 1 * 2 * 1 = 2. The minimum value for (b 3c) is 2 * 3 * 1 = 6. Continuing this pattern, the minimum value for (y 26z) is 25 * 26 * 1 = 650.
Finally, we have (z a), which means that the value of the expression is now 650 * 1 * 1 = 650.
To complete the expression, we need to divide it by abc...z. Since a, b, c, d, ..., z are positive integers, the minimum value for the product abc...z is 1 * 2 * 3 * ... * 26 = 26!.
Therefore, the minimum value of the expression is 650 / 26! ≈ 2.26.
So, the correct answer is a) 226.
First, we have (a 2b). This means that the value of the expression is a multiplied by 2b.
Next, we have (b 3c). This means that the value of the expression is now a multiplied by 2b, and then multiplied by 3c.
Continuing this pattern, we have (c 4d), (d 5e), and so on, until we reach (y 26z). Finally, we have (z a).
To find the minimum value of the expression, we want to minimize each term in parentheses. Since a, b, c, d, ..., z are positive integers, the minimum value for each term is when the smaller number is 1 and the larger number is the next consecutive integer.
So, the minimum value for (a 2b) is 1 * 2 * 1 = 2. The minimum value for (b 3c) is 2 * 3 * 1 = 6. Continuing this pattern, the minimum value for (y 26z) is 25 * 26 * 1 = 650.
Finally, we have (z a), which means that the value of the expression is now 650 * 1 * 1 = 650.
To complete the expression, we need to divide it by abc...z. Since a, b, c, d, ..., z are positive integers, the minimum value for the product abc...z is 1 * 2 * 3 * ... * 26 = 26!.
Therefore, the minimum value of the expression is 650 / 26! ≈ 2.26.
So, the correct answer is a) 226.
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Consider the expression(a + 2b) (b + 3c) (c + 4d) .........(y + 26z) (z + a) / abc...zWhere a, b, c, d , ...... z are positive integers. The minimum value of the expression is a)226√27! - 1b)226√26!c)227√26!d)27!e)227√27!Correct answer is option 'B'. Can you explain this answer?
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