if three numbers are in G.P. then middle one is said to be geometric mean (GM) of two others.
Let, G be the geometric mean between two number a and b
So, a G b are in G.P.
G/a = b/G.
or, G2 = ab
∴ G =√ab
Similarly we can find two geometric means between two given numbers a and b.
Let a, G1, G2, b are in G.P.
tn = a rn–1
or b/a = r3
r =(b/a)1/3
G1 = ar2 = a (b/a)1/3 = a1/3 b2/3
Geometric Mean(s)
• If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P., then b = √ac is the geometric mean of a and c.
• If a1, a2, ……, an are non-zero positive numbers, then their G.M.(G) is given by G = (a1a2a3, ……, an)1/n. If G1, G2, …… Gn are n geometric means between and a and b then a, G1, G2, ……, Gn b will be a G.P. Here b = arn+1.
⇒ r = n+1√b/a ⇒ G1 = an+1√b/a, G2 = a(n+1√b/a)2,…, Gn = a(n+1√b/a)n.
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