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In an arithmetic progression, if the terms at two different positions are equal, i.e. \(a_p=a_q\) where \(p\ne q\), which conclusion is necessary?किसी समांतर श्रेणी में यदि दो भिन्न स्थानों के पद समान हों, अर्थात् \(a_p=a_q\) जहाँ \(p\ne q\), तो कौन-सा निष्कर्ष अनिवार्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: The common difference is \(0\), so all terms are equalसार्वांतर \(0\) है और सभी पद समान हैं
Using \(a_p=a+(p-1)d\) and \(a_q=a+(q-1)d\) gives \((p-q)d=0\). Since \(p\ne q\), \(d=0\), so every term is equal. Exam tip: isolate the difference of indices first.\(a_p=a+(p-1)d\) और \(a_q=a+(q-1)d\) रखने पर \((p-q)d=0\) मिलता है। चूँकि \(p\ne q\), इसलिए \(d=0\) होगा और हर पद समान होगा। परीक्षा टिप: भिन्न सूचकांकों को पहले अलग करें।
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