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Misc 21 - Chapter 9 Class 11 Sequences and Series - Part 5

Misc 21 - Chapter 9 Class 11 Sequences and Series - Part 6
Misc 21 - Chapter 9 Class 11 Sequences and Series - Part 7 Misc 21 - Chapter 9 Class 11 Sequences and Series - Part 8

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Misc 21 Find the sum of the following series upto n terms: (ii) .6 +.66 +. 666 +… .6 +.66 +. 666 +… to n terms Here, 0.66/0.6 = (66/100)/(6/10) = 1.1 & 0.666/0.66 = (666/1000)/(66/100) = 1.009 Thus, (π‘†π‘’π‘π‘œπ‘›π‘‘ π‘‘π‘’π‘Ÿπ‘š)/(πΉπ‘–π‘Ÿπ‘ π‘‘ π‘‘π‘’π‘Ÿπ‘š) β‰  (π‘‡β„Žπ‘–π‘Ÿπ‘‘ π‘‘π‘’π‘Ÿπ‘š)/(π‘†π‘’π‘π‘œπ‘›π‘‘ π‘‘π‘’π‘Ÿπ‘š) i.e. common ratio is not same ∴ This is not a GP We need to find sum Sum = 0.6 + 0.66 + 0.666 + … to n terms = 6 [0.1 + 0.11 + 0.111 +… to n terms ] = 6 [0.1 + 0.11 + 0.111 +… to n terms ] Multiplying & dividing by 9 = 6/9 Γ— 9[0.1 + 0.11 + 0.111 +… to n terms ] = 2/3 [0.9 + 0.99 + 0.999 +… to n terms ] = 2/3 [ (9/10) + (99/100)+ (999/1000)+… to n terms ] = 2/3 [((10 βˆ’ 1)/10) + ((100 βˆ’ 1)/100)+ ((1000 βˆ’ 1)/1000)+ …to n terms ] = 2/3[(1 βˆ’ 1/10)+ (1 βˆ’ 1/100)+ (1 βˆ’ 1/1000) + … to n terms ] = 2/3 [(1 + 1 +…….to n terms) βˆ’ (1/10 " + " 1/100 " + " 1/1000 " + … to n terms" )] = 2/3 [n Γ— 1 – (1/10 " + " 1/100 " + " 1/1000 " + … to n terms" )] Now, a = 1/10, r = 1/10 For, r < 1 i.e. Sn = (a(1 βˆ’π‘Ÿ^𝑛))/(1 βˆ’ π‘Ÿ) Putting value of a = 1/10 & r = 1/10 = (1/10 (1 βˆ’(1/10)^𝑛 ))/(1 βˆ’ 1/10) = (1/10 (1 βˆ’(1/10)^𝑛 ))/(9/10) = 1/10[1 – (10)^(βˆ’π‘›)] Γ— 10/9 = 1/9[1 – (10)^(βˆ’π‘›)] Thus, (1/10 " + " 1/100 " + " 1/1000 " + … to n terms" ) = 1/9[1 – (10)^(βˆ’π‘›)] Now, Sum = 2/3 [n – (1/10 " + " 1/102 " + " 1/103 " + … to n terms" )] Substitute (1/10 " + " 1/100 " + " 1/1000 " + … to n terms" ) = 1/9[1 – (10)^(βˆ’π‘›)] in (1) = 2/3 [n – 1/9[1 – (10)^(βˆ’π‘›)] = 2/3n – 2/3 Γ—1/9[1 – (10)^(βˆ’π‘›)] = 2/3 n – 2/27[1 – (10)^(βˆ’π‘›)] Hence , 0.6 + 0.66 + 0.666 + … to n terms = 2/3 n – 2/27[1 – (10)^(βˆ’π‘›)]

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.