Question

If a1 is the first of n geometric means between a and b, show that a1^n+1=a^n b

Answer 1

Inserting  $n$ geometric means between $a$ and $b$ we get the following GP series of total $(n+2)$ terms

$$a,a_1,a_2,...a_n,b$$

Let $r$ be the common ratio of this GP then $b$ becomes the $(n+2)th$ term of the series. So we have

$$a_1=ar,..., b=ar^{n+1}$$

Then

$$a_1^{n+1}=(ar)^{n+1}=a^n ar^{n+1}=a^nb$$

Therefore

$$a_1^{n+1}=a^nb$$