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 , 7 2 , 1 0 8 a n d .

Q2:

Find the geometric mean of ( 𝑎 1 9 ) and ( 𝑎 + 1 9 ) .

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

Q3:

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

Q4:

Find the geometric means of the sequence ( 2 , , , , 4 8 0 2 ) .

  • A 1 4 , 9 8 , 6 8 6 , 4 8 0 2 or 1 4 , 9 8 , 6 8 6 , 4 8 0 2
  • B 1 6 , 1 2 8 , 1 0 2 4 or 1 6 , 1 2 8 , 1 0 2 4
  • C 1 6 , 1 2 8 , 1 0 2 4 , 8 1 9 2 or 1 6 , 1 2 8 , 1 0 2 4 , 8 1 9 2
  • D 1 4 , 9 8 , 6 8 6 or 1 4 , 9 8 , 6 8 6

Q5:

Insert five positive geometric means between 2 1 3 8 and 6 7 2 1 9 .

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

Q6:

Insert four geometric means between 1 4 and 256.

  • A 1 , 1 4 , 1 1 6 , 1 6 4
  • B 1 , 4 , 1 6 , 6 4
  • C 1 , 1 4 , 1 1 6 , 1 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, 92
  • B100, 196
  • C196, 292
  • D4, 100
  • E20, 500

Q10:

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

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

Q11:

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

  • A36, 121
  • B2, 21
  • C2, 83
  • D36, 49
  • E49, 134

Q12:

Find the geometric mean of 9 𝑥 3 6 and 3 6 𝑦 4 0 .

  • A 1 8 𝑥 𝑦 1 8 1 0
  • B 1 8 𝑥 𝑦 9 1 0
  • C 1 8 𝑥 𝑦 9 2 0
  • D 1 8 𝑥 𝑦 1 8 2 0

Q13:

Find the geometric mean of 𝑥 𝑦 2 2 and 𝑥 𝑦 𝑥 + 𝑦 .

  • A | | 𝑥 𝑦 | | 2 2
  • B | 𝑥 + 𝑦 |
  • C 𝑥 + 𝑦 2 2
  • D | 𝑥 𝑦 |

Q14:

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

Q15:

Find the geometric mean of ( 𝑥 + 𝑦 ) 2 and ( 𝑥 𝑦 ) 2 .

  • A | 𝑥 𝑦 |
  • B 𝑥 + 𝑦 2 2
  • C | 𝑥 + 𝑦 |
  • D | 𝑥 𝑦 | 2 2

Q16:

Find the geometric mean of 𝑥 3 and 9 𝑥 𝑦 3 1 6 .

  • A 3 | | 𝑦 | | 𝑥 3 8
  • B 9 | | 𝑥 | | 𝑦 3 8
  • C 9 | | 𝑦 | | 𝑥 3 8
  • D 3 | | 𝑥 | | 𝑦 3 8

Q17:

Find the geometric mean of 3 2 and 6 2 .

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 1 6 and 216.

Q21:

Find the geometric mean of 29.75 and 42.84.

Q22:

Find the geometric means of the sequence ( 4 , , , , 1 0 2 4 ) .

  • A 1 6 , 6 4 , 2 5 6 , 1 0 2 4 or 1 6 , 6 4 , 2 5 6 , 1 0 2 4
  • B 2 0 , 1 0 0 , 5 0 0 or 2 0 , 1 0 0 , 5 0 0
  • C 2 0 , 1 0 0 , 5 0 0 , 2 5 0 0 or 2 0 , 1 0 0 , 5 0 0 , 2 5 0 0
  • D 1 6 , 6 4 , 2 5 6 or 1 6 , 6 4 , 2 5 6

Q23:

Insert three positive geometric means between 1 and 8 1 1 6 .

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

Q24:

Insert two geometric means between 2 9 and 6.

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

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