Insertion of n geometric mean

May 02, 2022, 16:45 IST

Insertion of N Geometric Mean

Definition:- 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 g₁, g₂, ………g_n are n geometric means between and a and b then a, g₁, g₂, ………, g_n b will be a g.p.

Formula description: -

Let a₁, g₂, g₃, g₄……….g_n be n geometric means between two given numbers a and b. then a, g₁, g₂, ………G_n,b will be in geometric progression.

So, b = (n+2)^th term of the geometric progression.

Then here r is the common ratio

b = a* rⁿ⁺¹

rⁿ⁺¹ = b/a

r = (b/a)^1/(n+1)

Example 1:- insert 4 geometric means between 3 and 96

Ans :- Insertion of n geometric mean

Let G₁, G₂, G₃, G₄ be the required geometric means.

Then, 3, G₁, G₂, G₃, G₄, 96 are in G.P.

Let r be the common ratio.

96 is the 6^th term.

Example 2 :- Insert three numbers between 1 and 256 so that the resulting sequence is a g.p.

Ans:- Let G₁, G₂, G₃ be three numbers between 1 and 256 such that 1, G₁, G₂, G₃, 256 is a G.P.

Therefore 256 = r⁴ giving r = 4 (Taking real roots only)

For r = 4, we have G₁ = ar = 4, G₂ = ar₂ = 16, G₃ = ar₃ = 64

Similarly, for r = –4, numbers are –4, 16 and –64.

Hence, we can insert 4, 16, 64 between 1 and 256 so that the resulting are in G.P

Related Topics

• Insertion of n arthmetic mean in given two numbers
• General Terms of Arithmetic Progression