Misc 16 - If a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are AP

Misc 16 If a(1/(𝑏 ) " + " 1/𝑐), b(1/𝑐 " + " 1/𝑎), c(1/𝑎 " + " 1/𝑏) are in AP., prove that a, b, c are in AP Given that a(1/(𝑏 ) " + " 1/𝑐), b(1/𝑐 " + " 1/𝑎), c(1/𝑎 " + " 1/𝑏) are in AP. If a(1/(𝑏 ) " + " 1/𝑐), b(1/𝑐 " + " 1/𝑎), c(1/𝑎 " + " 1/𝑏) are in AP Adding 1 to each term a(1/(𝑏 ) " + " 1/𝑐) + 1, b(1/𝑐 " + " 1/𝑎) + 1, c(1/𝑎 " + " 1/𝑏) + 1 are in AP a(1/(𝑏 ) " + " 1/𝑐) + 𝑎/𝑎 , b(1/𝑐 " + " 1/𝑎) + 𝑏/𝑏 , c(1/𝑎 " + " 1/𝑏) + 𝑐/𝑐 are in AP a (1/(𝑏 ) " + " 1/𝑐 " + " 1/𝑎) , b(1/𝑐 " + " 1/𝑎 " + " 1/(𝑏 )) , c(1/𝑎 " + " 1/(𝑏 ) " + " 1/𝑐) are in AP Divide each term by (1/𝑎 " + " 1/(𝑏 ) " + " 1/𝑐) 𝑎(1/(𝑏 ) " + " 1/𝑐 " + " 1/𝑎)/(1/𝑎 " + " 1/(𝑏 ) " + " 1/𝑐), 𝑏(1/𝑐 " + " 1/𝑎 " + " 1/(𝑏 ))/(1/𝑎 " + " 1/(𝑏 ) " + " 1/𝑐), 𝑐(1/𝑎 " + " 1/(𝑏 ) " + " 1/𝑐)/(1/𝑎 " + " 1/(𝑏 ) " + " 1/𝑐) are in AP a, b, c are AP Hence, a, b, c, are in AP Hence proved