1. Chapter 9 Class 11 Sequences and Series
2. Serial order wise
3. Miscellaneous

Transcript

Misc 13 If , (a+bx)/(aโbx) = (b+cx)/(bโcx) = (c+dx)/(aโdx) (x โ  0)then show that a, b, c and d are in G.P. Introduction Componendo dividendo If ๐ฅ/๐ฆ = ๐/๐ Applying componendo dividendo (๐ฅ + ๐ฆ)/(๐ฅ โ ๐ฆ) = (๐ + ๐)/(๐ โ ๐) Eg: Taking 1/2 = 4/8 (1+ 2)/(1 โ 2) = (4 + 8)/(4 โ 8) 3/(โ1) = 12/(โ4) -3 = -3 Misc 13 If , (a+bx)/(aโbx) = (b+cx)/(bโcx) = (c+dx)/(aโdx) (x โ  0)then show that a, b, c and d are in G.P. We have (a+bx)/(aโbx) = (b+cx)/(bโcx) = (c+dx)/(c โ dx) & we want to show that a, b, c, d are in G.P. Taking (a+bx)/(aโbx) = (b+cx)/(bโcx) = (c+dx)/(c โ dx) Applying componendo dividendo (a + bx + a โ bx)/((a + bx) โ(aโbx)) = (b + cx + (b โ cx))/(b + cx โ(b โ cx)) = (c + dx + (c โ dx))/(c + dx โ (c โ dx)) (a + a + bx โ bx)/(๐๐ฅ+ bx โ a + a ) = (b + b + cx โ cx)/(cx + cx โ ๐ + ๐) = (c + dx + c โ dx)/(dx + dx โ ๐ + ๐) (2๐+0)/(2๐๐ฅ+0) = (2๐ + 0)/(2๐๐ฅ + 0) = (2๐+0)/(2๐๐ฅ+0) 2๐/2๐๐ฅ = 2๐/2๐๐ฅ = 2๐/2๐๐ฅ ๐/๐๐ฅ = ๐/๐๐ฅ = ๐/๐๐ฅ a/b " =" b/c = c/d b/a " =" c/b = d/c Thus, a, b, c & d are in GP because their common ratio is same

Miscellaneous