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.
हिंदी में प्रश्न एवं आदर्श उत्तर
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।