अनुक्रम 14, 18, 20, 24, 30, 32, ... की प्रत्येक संख्या में एक सरल गणितीय संक्रिया द्वारा अभाज्य संख्याओं का एक अनुक्रम प्राप्त किया जाता है। निम्न में से कौन-सी अनुक्रम की अगली संख्या है?
सही उत्तर: (c) 38
दी गई श्रृंखला 14, 18, 20, 24, 30, 32, ... में प्रत्येक संख्या में से 1 घटाने पर क्रमशः अभाज्य संख्याएँ प्राप्त होती हैं: 14-1=13, 18-1=17, 20-1=19, 24-1=23, 30-1=29, 32-1=31। ये सभी क्रमिक अभाज्य संख्याएँ हैं। 31 के बाद अगली अभाज्य संख्या 37 है, अतः अगली संख्या x के लिए x - 1 = 37, जिससे x = 38 प्राप्त होता है।
In English (Question & Model Answer)
A simple mathematical operation in each number of the sequence 14, 18, 20, 24, 30, 32, ... results in a sequence with respect to prime numbers. Which one of the following is the next number in the sequence?
Correct Option: (c) 38
Let's examine the sequence: 14, 18, 20, 24, 30, 32, ... Apply a simple mathematical operation to each term to get a sequence of prime numbers. Let's check subtracting or adding a constant, or subtracting from consecutive primes. Let's look at prime numbers: 17, 19, 23, 29, 31, 37, ... Notice the relationship between the sequence numbers and primes: 17 - 3 = 14. 19 - 1 = 18? No. Let's try adding or subtracting: 14 + 3 = 17 (prime), 18 + 1 = 19 (prime), 20 + 3 = 23 (prime), 24 + 5 = 29 (prime), 30 + 1 = 31 (prime), 32 + 5 = 37 (prime). Look at the added numbers: 3, 1, 3, 5, 1, 5... Wait, let's check another operation: subtract 1? 14 - 1 = 13 (prime), 18 - 1 = 17 (prime), 20 - 1 = 19 (prime), 24 - 1 = 23 (prime), 30 - 1 = 29 (prime), 32 - 1 = 31 (prime). Wow! If we subtract 1 from each term, we get: 13, 17, 19, 23, 29, 31. These are consecutive prime numbers in ascending order! The next prime number after 31 is 37. Therefore, if x - 1 = 37, then x = 37 + 1 = 38. Thus, the next number in the sequence is 38.