## Two cute problems in HP : IITJEE Foundations\Mains, pre RMO

**Problem 1: **

If are in AP, show that are in HP.

**Proof 1:**

Note that a straight forward proof is not so easy.

Below is a nice clever solution:

By adding to each term, we see that:

are in AP.

that is, are in AP.

Dividing each term by .

are in AP.

that is, are in HP.

QED.

**Problem 2:**

If the terms of an AP are in GP, show that are in GP.

**Proof 2:**

Once again a straight forward proof is not at all easy.

Below is a “bingo” sort of proof 🙂

With the usual notation, we have

Hence, each of the ratios is equal to

which in turn is equal to

Hence, are in GP.

Cheers,

Nalin Pithwa

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