Q no 2 Q2 Show that a1a2+a2a3+a3a4 + an-1an+ana1 n, where a1, a2 - Maths - Sequences and Series

Question

Q no 2

Q2 Show that a 1 a 2 + a 2 a 3 + a 3 a 4 + a n - 1 a n + a n a 1
  > n ,   where   a 1 ,   a 2   an are
different positive integers

Aarushi Mishra · Accepted Answer

Consider n positive integers a1a2,a2a3,a3a4,
,an-1an and ana1Since all integer are positive, we can apply A
M  and G M  inequalityA M ≥G M a1a2+a2a3+a3a4+
+an-1an+ana1n≥a1a2×a2a3×a3a4×an-1an×ana11na1a2+a2a3+a3a4+
+an-1an+ana1n≥a1a2×a2a3×a3a4×an-1an×ana11na1a2+a2a3+a3a4+
+an-1an+ana1n≥11na1a2+a2a3+a3a4+
+an-1an+ana1n≥1a1a2+a2a3+a3a4+ +an-1an+ana1≥n

Q no 2 Q2. Show that a 1 a 2 + a 2 a 3 + a 3 a 4 ... + a n - 1 a n + a n a 1 > n , where a 1 , a 2 . . . an are different positive integers.

Q no 2

Q2. Show that $\frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}} + \frac{a_{3}}{a_{4}}$ ... + $\frac{a_{n - 1}}{a_{n}} + \frac{a_{n}}{a_{1}} > n, where a 1, a 2 . . . an$ are different positive integers.