Manipal Entrance Test 2020 Math Paper

Given that, a,b and c are in HP.
∴ Harmonic mean of a and c is b.
and geometric mean of a and c is √ac.
Geometric mean > Harmonic mean
⇒‌‌√ac>b......(i)
Now, for the positive numbers an and cn, we have
Geometric mean =√an⋅cn
and arithmetic mean =‌

an+cn

2

∵‌‌AM>GM
∴‌‌‌

an+cn

2

>√an⋅cn
⇒‌‌an+cn>2√an⋅cn
⇒‌‌an+cn>2(√ac)n>2bn [using Eq. (i)]
⇒‌‌an+cn>2bn