Worksheet: Finding the nth Term of a Geometric Sequence

In this worksheet, we will practice writing explicit and recursive formulas for geometric sequences to find the value of the nth term in a geometric sequence and how to find a term's order given its value.

Q1:

Find the first three terms of a geometric sequence given 𝑎=3,616 and the common ratio is 2.

  • A 1 1 1 3 , 2 1 1 3 , 4 1 1 3
  • B 1 1 3 , 2 2 6 , 4 5 2
  • C 1 1 3 , 2 2 6 , 4 5 2
  • D 1 1 3 , 1 1 3 2 , 1 1 3 4
  • E 1 1 3 , 1 1 3 2 , 1 1 3 4

Q2:

Find the value of 𝑎 in a geometric sequence given 6𝑎+𝑎=16𝑎,𝑎=62 and all terms are positive.

  • A 3 1 6 4
  • B 3 1 3 2
  • C 3 1 4
  • D 3 1 2 5 6
  • E 3 1 1 2 8

Q3:

Find the three consecutive numbers of a geometric sequence, given that the sum of the terms is 14 and the product is 216.

  • A 2 , 6, 18
  • B 1 6 , 1 1 8 , 1 5 4
  • C 1 2 , 1 6 , 1 1 8
  • D6, 18, 54
  • E6, 2, 23

Q4:

Find the three consecutive numbers of a geometric sequence, given their sum is 7 and the product of their squares is 64.

  • A2, 1, 12
  • B 1 2 , 1, 2
  • C 1 4 , 1 2 , 1
  • D2, 4, 8
  • E4, 2, 1

Q5:

Find the first three terms of the geometric sequence given their sum is 7 and the sum of their squares is 21.

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

Q6:

Find the order of the term 1639 in the geometric sequence 1156,178,139,.

Q7:

Find the second and third terms of a geometric sequence given 𝑎=69,𝑎=4,416, and all terms are positive.

  • A 𝑎 = 6 9 2 , 𝑎 = 6 9 4
  • B 𝑎 = 1 3 8 , 𝑎 = 2 7 6
  • C 𝑎 = 6 9 4 , 𝑎 = 6 9 8
  • D 𝑎 = 1 3 8 , 𝑎 = 5 5 2
  • E 𝑎 = 2 7 6 , 𝑎 = 5 5 2

Q8:

The numbers 38 and 4,864 are terms of a geometric sequence. The ratio of the sum of the two terms after 38 in the sequence to the sum of the two terms before 4,864 in the sequence is 116. How many terms lie between 38 and 4,864 in the sequence?

Q9:

In a geometric sequence and for a certain 𝑛, we have 𝑎=6, 𝑎=3, and 𝑎=38. Find the first 3 terms of the sequence and what 𝑛 is.

  • A 9 6 , 4 8 , 2 4 , and 𝑛=4
  • B 1 9 2 , 9 6 , 4 8 , and 𝑛=5
  • C 3 8 , 3 4 , 3 2 , and 𝑛=5
  • D 3 1 6 , 3 8 , 3 4 , and 𝑛=5
  • E 9 6 , 4 8 , 2 4 , and 𝑛=5

Q10:

Find, in terms of 𝑛, the general term of the sequence (1,2,4,8,) where 𝑛1.

  • A ( 2 )
  • B 2 𝑛 1
  • C 2
  • D 2 1

Q11:

Determine 𝑝 and 𝑞 that make the following a geometric sequence, starting at 𝑎, of the form 𝑎=𝑝𝑞: 2,6,18,54,162,

  • A 𝑝 = 4 , 𝑞 = 3
  • B 𝑝 = 3 , 𝑞 = 2
  • C 𝑝 = 2 , 𝑞 = 2
  • D 𝑝 = 2 , 𝑞 = 3
  • E 𝑝 = 3 , 𝑞 = 3

Q12:

Find the value of 𝑛 given the 𝑎 of the geometric sequence (96,24,6,) is three times greater than the 𝑎 of the geometric sequence (4,2,1,).

Q13:

The sum of the first and second terms of a geometric sequence is 54 and 𝑎+𝑎=216. Find the possible values of 𝑎.

  • A 9 2 or 272
  • B144 or 432
  • C72 or 216
  • D9 or 27

Q14:

Find, in terms of 𝑛, the general term of the sequence (1,3,9,27,81,).

  • A 3
  • B 3 𝑛
  • C 3
  • D 3 ( 𝑛 1 )

Q15:

Find the order of the term whose value is 4,374 in the geometric sequence 𝑎=23(3).

Q16:

Find the order and value of the first term whose value is less than 0.0049 in the geometric sequence (2,0.4,0.08,).

  • A 𝑛 = 6 , 𝑎 = 2 3 , 1 2 5
  • B 𝑛 = 4 , 𝑎 = 2 6 2 5
  • C 𝑛 = 4 , 𝑎 = 2 1 2 5
  • D 𝑛 = 5 , 𝑎 = 2 6 2 5
  • E 𝑛 = 5 , 𝑎 = 2 3 , 1 2 5

Q17:

Suppose that 𝑎=5 and 𝑎=3 belong to a geometric sequence with a positive 𝑎. Using four decimal places for your common ratio, determine this first term to two decimal places.

  • A 𝑎 = 1 3 . 8 9
  • B 𝑎 = 9 . 0 0
  • C 𝑎 = 0 . 7 7
  • D 𝑎 = 6 4 . 3 0
  • E 𝑎 = 1 . 8 0

Q18:

Find the number of terms of the geometric sequence 112,56,28,,74.

Q19:

Find the order of the first term whose value is less than one in the geometric sequence (1,029,588,336,).

Q20:

Find the second term of a geometric sequence given 𝑎𝑎=64, 𝑎𝑎=48 and all terms are positive.

Q21:

Find the geometric sequence given the first term is 3 and the fourth term is 81. Then find the order of the term whose value is 729.

  • A 3 , 1 , 1 3 , , 6
  • B 3 , 1 , 1 3 , , 5
  • C ( 3 , 9 , 2 7 , ) , 5
  • D ( 3 , 9 , 2 7 , ) , 6
  • E ( 3 , 9 , 2 7 , ) , 6

Q22:

What is the order of the term in the geometric sequence 6,24,96, whose value is 1,572,864?

Q23:

Find, in terms of 𝑛, the general term of the sequence 14,916,8164,729256,.

  • A 9 4
  • B 9 4
  • C 9 4
  • D 9 4

Q24:

Find the geometric sequence given 𝑎=4,096,𝑎=64 and the common ratio is 4.

  • A 4 , 4 , 4 ,
  • B 4 , 4 , 4 ,
  • C 4 , 4 , 4 ,
  • D 4 , 4 , 4 ,
  • E 4 , 4 , 4 ,

Q25:

Determine 𝑝 and 𝑞 that make 5,157,4549,135343,4052,401, a geometric sequence, starting at 𝑎, of the form 𝑎=𝑝𝑞.

  • A 𝑝 = 3 , 𝑞 = 5
  • B 𝑝 = 2 , 𝑞 = 1 7
  • C 𝑝 = 7 , 𝑞 = 5 7
  • D 𝑝 = 5 , 𝑞 = 3 7
  • E 𝑝 = 5 , 𝑞 = 3 7

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