📌 Example 1.
Insert two numbers between 3 and 81 so that the resulting sequence is geometric sequence.
✍ Solution:
Let G₁ and G₂ be two numbers between 3 and 81 such that the series, 3, G₁, G₂, 81, forms a geometric sequence.
Let a be the first term and r be the common ratio of the geometric sequence.

∴ 81=3r³
r³=27
r=3
G₁=a₂=ar=3⋅3=9,
G₂=a₃=ar²=3⋅3²=27.
Thus, the required two numbers are 9 and 27.

📌 Ex2. Find the indicated geometric means between two terms.
📌 Ex2a. 0.20, _, _, _, 125
✍ Solution: Given a₁=0.20 and a₅=125.

a_n=a₁⋅r^(n-1)
125=0.20⋅r^(5-1)
r⁴=625
r=±5
The geometric means are 1, 5, 25 or -1, 5, -25.
Answer: 1, 5, 25 or -1, 5, -25

📌 Ex2b. 4 and 256; 2 means
✍ Solution:

The sequence will resemble 4, ? , ? , 256.

Note that a₁=4, n=4, and a₄=256. Find the common ratio using nth term for a geometric sequence formula.

a_n=a₁⋅r^(n-1)
256=4r³
64=r³
4=r
The common ratio is 4. Use r to find the geometric means.
4(4)=16
16(4)=64
Therefore, a sequence with two geometric means between 4 and 256 is 4, 16, 64, 256.

📌 Ex2c. 256 and 81; 3 means
✍ Solution:
The sequence will resemble 256, ? , ? , ? , 81. Note that a₁=256, n=5, and a₅=81.
Find the common ratio using the nth term for a geometric sequence formula.

The common ratio is ±¾. Use r to find the geometric means.
r=-¾
256⋅(-¾)=-192
-192⋅(-¾)=144
144⋅(-¾)=108

r=¾
256⋅(¾)=192
192⋅(¾)=144
144⋅(¾)=108
Therefore, a sequence with three geometric means between 256 and 81 is 256, -192, 144, -108, 81 or 256, 192, 144, 108, 81.

📌 Ex2d. 4/7 and 7; 1 mean
✍ Solution:
The sequence will resemble 4/7, ? , 7. Note that a₁=4/7, n=3, and a₃=7. Find the common ratio using the nth term for a geometric sequence formula.

The common ratio is ±7/2 . Use r to find the geometric means.

Therefore, a sequence with one geometric mean between 4/7 and 7 is 4/7, -2 , 7 or 4/7, 2 , 7

📌 Ex2e. -2 and 54; 2 means
✍ Solution:
The sequence will resemble -2, ? , ? , 54.
Note that a₁=-2, n=4, and a₄=54. Find the common ratio using nth term for a geometric sequence formula.

a_n=a₁⋅r^(n-1)
54=-2r³
-27=r³
-3=r
The common ratio is -3. Use r to find the geometric means.
-2(-3)=6
6(-3)=-18
Therefore, a sequence with two geometric means between -2 and 54 is -2, 6, -18, 54.

📌 Ex2f. 1 and 27; 2 means
✍ Solution: The sequence will resemble 1, ? , ? , 27.
Note that a₁=1, n=4, and a₄=27. Find the common ratio using nth term for a geometric sequence formula.

a_n=a₁⋅r^(n-1)
27=1r³
3³=r³
3=r
The common ratio is 3. Use r to find the geometric means.
1(3)=3
3(3)=9
Therefore, a sequence with two geometric means between 1 and 27 is 1, 3, 9, 27.

📌 Ex2g. 48 and -750; 2 means
✍ Solution:

The sequence will resemble 48, ? , ? , -750.
Note that a₁=48, n=4, and a₄=48. Find the common ratio using nth term for a geometric sequence formula.

The common ratio is -5/2. Use r to find the geometric means.

Therefore, a sequence with two geometric means between 48 and -750 is 48, -120, 300, -750.

📌 Ex2h. t⁸ and t^(-7); 4 means
✍ Solution:
The sequence will resemble t⁸, ? , ? , ? , ? , t^(-7).
Note that a₁=t⁸, n=6, and a₆=t^(-7). Find the common ratio using nth term for a geometric sequence formula.

The common ratio is t^(-3). Use r to find the geometric means.
t⁸⋅t^(-3)=t⁵
t⁵⋅t^(-3)=t²
t²⋅t^(-3)=t^(-1)
t^(-1)⋅t^(-3)=t^(-4)
Therefore, a sequence with two geometric means between t⁸ and t^(-7) is t⁸, t⁵, t², t^(-1), t^(-4), t^(-7).