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समांतर श्रेणी

Question
CBSEHHIMAH10010278

एक पंक्ति के मकानों को क्रमागत रुप से संख्या 1 से 49 तक अंकित किया गया है। दर्शाइए कि x का एक ऐसा मान है कि x से अंकित मकान से पहले के मकानों कि संख्याओं का योग उसके बाद वाले मकानों की संख्याओं के योग बराबर है। x का मान ज्ञात कीजिए।
[ संकेत: Sx - 1 = S49 - Sx है ]

Solution

माना मकानों के नंबर है: 1, 2, 3, ... 49 जो कि एक A.P. में हैं।
यहाँ, a = 1, d = 2 – 1 = 1
प्रश्नानुसार
straight S subscript straight x space minus space 1 end subscript space equals space straight S subscript 49 space minus space straight S subscript straight x
rightwards double arrow space fraction numerator straight x space minus space 1 over denominator 2 end fraction space left square bracket space 2 straight a space plus space left parenthesis space straight x space minus space 1 space minus space 1 space right parenthesis space straight d space right square bracket
equals space 49 over 2 space left square bracket space 2 straight a space plus space left parenthesis space 49 space minus space 1 space right parenthesis space straight d space right square bracket space minus space straight x over 2 space left square bracket space 2 straight a space plus space left parenthesis space straight x space minus space 1 space right parenthesis space 1 space right square bracket
open square brackets space because space space straight S subscript straight n space equals space straight n over 2 space left square bracket space 2 straight a space plus space left parenthesis space straight n space minus space 1 space right parenthesis space straight d space right square bracket space close square brackets
rightwards double arrow space fraction numerator straight x space minus 1 over denominator 2 end fraction space left square bracket space 2 space left parenthesis space 1 space right parenthesis space plus space left parenthesis space straight x space minus space 2 space right parenthesis space left parenthesis space 1 space right parenthesis space right square bracket
rightwards double arrow space 49 over 2 space left square bracket space 2 space left parenthesis space 1 space right parenthesis space plus space left parenthesis space 48 space right parenthesis space left parenthesis space 1 space right parenthesis space right square bracket space minus space straight x over 2 space left square bracket space 2 space left parenthesis space 1 space right parenthesis space plus space left parenthesis space straight x space minus space 1 space right parenthesis space straight d space right square bracket
rightwards double arrow space fraction numerator straight x space minus space 1 over denominator 2 end fraction space left square bracket space straight x space right square bracket space equals space 1225 space minus space fraction numerator straight x space left parenthesis space straight x space plus space 1 space right parenthesis over denominator 2 end fraction
rightwards double arrow space fraction numerator left parenthesis space straight x space minus space 1 space right parenthesis space left parenthesis space straight x space right parenthesis over denominator 2 end fraction space plus space fraction numerator straight x space left parenthesis space straight x space plus space 1 space right parenthesis over denominator 2 end fraction space equals space 1225
rightwards double arrow space straight x over 2 space left parenthesis space straight x space minus space 1 space plus space straight x space plus space 1 space right parenthesis space equals space 1225
rightwards double arrow space straight x squared space equals space 1225
rightwards double arrow space straight x space equals space square root of 1225
rightwards double arrow space straight x space equals space 35