Question
Download Solution PDFयदि C0, C1, C2, _ _ _ _ _, Cn, (1 + x)n के विस्तार में गुणांक हैं तो C1 + C2 + C3 + _ _ _ _ _ + Cn का मान क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFअवधारणा:
(1 + x)n = C0 + x. C1 + x2. C2 + ...... + xnCn
गणना:
माना S = C1 + C2 + C3 + _ _ _ _ _ + Cn
⇒ S = nC1 + nC2 + nC3 + _ _ _ _ _ + nCn
⇒ S = 1 + nC1 + nC2 + nC3 + _ _ _ _ _ + nCn - 1
⇒ S = nC0 + nC1 + nC2 + nC3 + _ _ _ _ _ + nCn - 1 ----(1)
जैसा कि हम जानते हैं, (1 + x)n = C0 + x. C1 + x2. C2 + ...... + x nCn
x = 1 रखने पर, हम प्राप्त करते है
⇒ (1 + x)n = nC0 + nC1 + nC2 + nC3 + .... + nCn = 2n -----(2)
समीकरण (1) और (2) से, हम प्राप्त करते हैं
S = nC0 + nC1 + nC2 + nC3 + _ _ _ _ _ + nCn - 1
∴ C1 + C2 + C3 + _ _ _ _ _ + Cn का मान 2n - 1 है
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