Last updated at May 29, 2018 by Teachoo

Transcript

Misc 17 If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P. We know that a, ar , ar2 , ar3, …. are in G.P. with first term a & common ratio r Given a, b, c, d are in G.P. So, a = a b = ar c = ar2 d = ar3 We want to show that (an + bn), (bn + cn), (cn + dn) are in GP i.e. to show common ratio are same (𝑏^𝑛+𝑐^𝑛)/(𝑎^𝑛+𝑏^𝑛 ) = (𝑐^𝑛+𝑑^𝑛)/(𝑏^𝑛+𝑐^𝑛 ) Need to show (𝑏^𝑛+𝑐^𝑛)/(𝑎^𝑛+𝑏^𝑛 ) = (𝑐^𝑛+𝑑^𝑛)/(𝑏^𝑛+𝑐^𝑛 ) Taking L.H.S (𝑏^𝑛+𝑐^𝑛)/(𝑎^𝑛+𝑏^𝑛 ) Putting b = ar , c = ar2 = (〖(𝑎𝑟)〗^𝑛+〖(〖𝑎𝑟〗^2)〗^𝑛)/(𝑎^𝑛+〖(𝑎𝑟)〗^𝑛 ) = (𝑎^𝑛 𝑟^𝑛 + 𝑎^𝑛.𝑟^2𝑛)/(𝑎^𝑛 + 𝑎^𝑛 𝑟^𝑛 ) = (𝑎^𝑛 𝑟^𝑛 (1 +𝑟^𝑛))/(𝑎^𝑛 (1 +𝑟^𝑛)) = rn Taking R.H.S (𝑐^𝑛+𝑑^𝑛)/(𝑏^𝑛+𝑐^𝑛 ) Putting c = ar2, d = ar3, b = ar = (〖(〖𝑎𝑟〗^2)〗^𝑛 + 〖(〖𝑎𝑟〗^3)〗^𝑛)/(〖(𝑎𝑟)〗^𝑛 +〖 (〖𝑎𝑟〗^2)〗^𝑛 ) = (𝑎^𝑛 𝑟^2𝑛+ 𝑎^𝑛 .𝑟^2𝑛)/(𝑎^𝑛 𝑟^𝑛+〖 𝑎〗^𝑛 𝑟^2𝑛 ) = (𝑎^𝑛 (𝑟^2𝑛 + 𝑟^3𝑛))/(𝑎^𝑛 (𝑟^𝑛 + 𝑟^2𝑛)) = ( (𝑟^2𝑛 + 𝑟^3𝑛))/((𝑟^𝑛 + 𝑟^2𝑛)) = (𝑟^2𝑛 (1 + 𝑟))/(𝑟^𝑛 (1 + 𝑟)) = 𝑟^2𝑛/𝑟^𝑛 = (𝑟^𝑛.𝑟^𝑛)/𝑟^𝑛 = rn = rn = L.H.S Thus L.H.S = R.H.S Hence (an + bn), (bn + cn) & (cn + bn) are in GP Hence proved

Miscellaneous

Misc 1

Misc 2

Misc 3 Important

Misc 4

Misc 5

Misc 6

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11

Misc 12

Misc 13

Misc 14

Misc 15

Misc 16 Important

Misc 17 You are here

Misc 18

Misc 19 Important

Misc 20

Misc 21

Misc 22

Misc 23

Misc 24

Misc 25 Important

Misc 26

Misc 27

Misc 28 Important

Misc 29

Misc 30

Misc 31

Misc 32 Important

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.