Worksheet: Geometric Mean

In this worksheet, we will practice finding geometric means between two nonconsecutive terms of a geometric sequence.

Q1:

Find the geometric mean of the numbers 6,72,108and.

Q2:

Find the geometric mean of (𝑎19) and (𝑎+19).

  • A 𝑎
  • B 𝑎 3 6 1
  • C 𝑎 1 9
  • D 𝑎 3 6 1
  • E 2 𝑎

Q3:

Find 𝑦 given the geometric mean between 2 and 𝑦 is 10.

Q4:

Find the geometric means of the sequence (2,,,,4,802).

  • A 1 4 , 9 8 , 6 8 6 or14,98,686
  • B 1 4 , 9 8 , 6 8 6 , 4 , 8 0 2 or14,98,686,4,802
  • C 1 6 , 1 2 8 , 1 , 0 2 4 , 8 , 1 9 2 or16,128,1,024,8,192
  • D 1 6 , 1 2 8 , 1 , 0 2 4 or16,128,1,024

Q5:

Insert five positive geometric means between 2138 and 67219.

  • A 2 1 7 6 , 2 1 1 5 2 , 2 1 3 0 4 , 2 1 6 0 8 , 2 1 1 , 2 1 6
  • B 2 1 1 9 , 4 2 1 9 , 8 4 1 9 , 1 6 8 1 9 , 3 3 6 1 9
  • C 2 1 1 9 , 4 2 1 9 , 8 4 1 9 , 1 6 8 1 9 , 3 3 6 1 9
  • D 2 1 7 6 , 2 1 1 5 2 , 2 1 3 0 4 , 2 1 6 0 8 , 2 1 1 , 2 1 6

Q6:

Insert four geometric means between 14 and 256.

  • A 1 , 1 4 , 1 1 6 , 1 6 4
  • B 1 , 1 4 , 1 1 6 , 1 6 4
  • C 1 , 4 , 1 6 , 6 4
  • D 1 , 4 , 1 6 , 6 4
  • E 1 2 , 1 , 2 , 4

Q7:

Find the number of geometric means inserted between 82 and 1,312 given the sum of the last two means equals twice the sum of the first two means.

Q8:

How many geometric means are inserted between 81 and 16 so that the product of the second mean and the last mean is 864?

Q9:

Find the two positive numbers whose positive geometric mean is greater than the smaller number by 16 and less than the larger number by 80.

  • A4, 100
  • B20, 500
  • C196, 292
  • D4, 92
  • E100, 196

Q10:

Find two positive numbers given their geometric mean is 36 and their difference is 21.

  • A 2 7 , 6
  • B 3 9 , 1 8
  • C 4 8 , 2 7
  • D 1 8 , 2

Q11:

Find two positive numbers given the geometric mean is 42 and the sum is 85.

  • A2, 21
  • B2, 83
  • C36, 49
  • D49, 134
  • E36, 121

Q12:

Find the geometric mean of 9𝑥 and 36𝑦.

  • A 1 8 𝑥 𝑦
  • B 1 8 𝑥 𝑦
  • C 1 8 𝑥 𝑦
  • D 1 8 𝑥 𝑦

Q13:

Find the geometric mean of 𝑥𝑦 and 𝑥𝑦𝑥+𝑦.

  • A | | 𝑥 𝑦 | |
  • B | 𝑥 + 𝑦 |
  • C 𝑥 + 𝑦
  • D | 𝑥 𝑦 |

Q14:

The ratio 𝑥4=4𝑦, so 4 is the geometric mean of 𝑥 and 𝑦. Find the geometric mean of 𝑥+1𝑦 and 𝑦+1𝑥.

Q15:

Find the geometric mean of (𝑥+𝑦) and (𝑥𝑦).

  • A | 𝑥 𝑦 |
  • B | 𝑥 𝑦 |
  • C | 𝑥 + 𝑦 |
  • D 𝑥 + 𝑦

Q16:

Find the geometric mean of 𝑥 and 9𝑥𝑦.

  • A 9 | | 𝑥 | | 𝑦
  • B 9 | | 𝑦 | | 𝑥
  • C 3 | | 𝑦 | | 𝑥
  • D 3 | | 𝑥 | | 𝑦

Q17:

Find the geometric mean of 3 and 6.

Q18:

The three sides of a right triangle are in a geometric sequence with a common ratio 𝑟<1. Find 𝑟.

  • A 5 1 2
  • Bcannot be determined
  • C 5 + 1 2
  • D 5 + 1 2
  • E 5 + 1 2

Q19:

Find the geometric mean of 16 and 4.

Q20:

Find the geometric mean of 16 and 216.

Q21:

Find the geometric mean of 29.75 and 42.84.

Q22:

Find the geometric means of the sequence (4,,,,1,024).

  • A 1 6 , 6 4 , 2 5 6 , 1 , 0 2 4 or16,64,256,1,024
  • B 2 0 , 1 0 0 , 5 0 0 or20,100,500
  • C 1 6 , 6 4 , 2 5 6 or16,64,256
  • D 2 0 , 1 0 0 , 5 0 0 , 2 , 5 0 0 or20,100,500,2,500

Q23:

Insert three positive geometric means between 1 and 8116.

  • A 3 2 , 9 4 , 2 7 8
  • B 3 2 , 9 4 , 2 7 8
  • C 2 3 , 4 9 , 8 2 7
  • D 2 3 , 4 9 , 8 2 7
  • E 6 , 3 6 , 2 1 6

Q24:

Insert two geometric means between 29 and 6.

  • A 2 3 , 2
  • B 3 2 , 1 2
  • C 1 3 , 1 2
  • D 2 3 , 2
  • E 3 2 , 1 2

Q25:

Is the arithmetic mean of two positive, different real numbers greater than their geometric mean?

  • Ano
  • Byes

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