Question
Download Solution PDF\(\displaystyle\sum_{r=0}^n \)2r C(n, r) किसके बराबर है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
(x + y)n का द्विपद प्रसरण निम्न द्वारा दिया जाता है
(x + y)n = nC0xn + nC1xn-1y + nC2xn-2y2+.....+ nCn-1xyn-1 + nCnyn
गणना:
दिया गया है, \(\displaystyle\sum_{r=0}^n\)2r C(n, r)
व्यंजक का प्रसार करने पर,
⇒ nC020 + nC1 21 + nC222+.....+ nCn-12n-1 + nCn2n
⇒ nC01n 20 + nC1 1n-1 21 + nC21n-2 22+.....+ nCn-1 2n-1 + nCn2n
द्विपद प्रसरण की तुलना करने पर x = 1 और y = 2
⇒ \(\displaystyle\sum_{r=0}^n\)2r C(n, r) = (1 + 2)n
⇒ \(\displaystyle\sum_{r=0}^n\)2r C(n, r) = 3n
∴ सही विकल्प (2) है।
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