How to Write all Terms (Geometric Means) Between Two Terms in a Geometric Sequence?

📌 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.

an=a1r(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.

an=a1r(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.

common ratio u

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

r
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.

common ratio v

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

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.

an=a1r(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.

an=a1r(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.

common ratio x

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

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.

common ratio z

The common ratio is t(-3). Use r to find the geometric means.
t8t(-3)=t5
t5t(-3)=t2
t2t(-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).

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