Lesson Worksheet: Finding the nth Term of a Geometric Sequence Mathematics

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.

  • A1113,−2113,4113
  • B113,226,452
  • C113,−226,452
  • D113,−1132,1134
  • E113,1132,1134

Q2:

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

  • A3164
  • B−3132
  • C314
  • D31256
  • E31128

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
  • B16, −118, 154
  • C−12, 16, −118
  • 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.

  • A14, 12, 1
  • B2, 1, 12
  • C4, 2, 1
  • D12, 1, 2
  • E2, 4, 8

Q5:

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

  • A−14,−12,−1
  • B−4,−2,−1
  • C−4,−8,−16
  • D−14,−18,−116

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.

  • A𝑎=692, 𝑎=694
  • B𝑎=138, 𝑎=276
  • C𝑎=694, 𝑎=698
  • D𝑎=138, 𝑎=552
  • E𝑎=276, 𝑎=552

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 1∶16. 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.

  • A96,48,24, and 𝑛=4
  • B192,96,48, and 𝑛=5
  • C38,34,32, and 𝑛=5
  • D316,38,34, and 𝑛=5
  • E96,48,24, and 𝑛=5

Q10:

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

  • A(2)
  • B2𝑛−1
  • C2
  • D2−1

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,…

  • A𝑝=4, 𝑞=3
  • B𝑝=3, 𝑞=2
  • C𝑝=2, 𝑞=2
  • D𝑝=2, 𝑞=3
  • E𝑝=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.

  • A92 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,…).

  • A3
  • B3𝑛
  • C3
  • D3(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,…).

  • A𝑛=6, 𝑎=23,125
  • B𝑛=4, 𝑎=2625
  • C𝑛=4, 𝑎=2125
  • D𝑛=5, 𝑎=2625
  • E𝑛=5, 𝑎=23,125

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.

  • A𝑎=13.89
  • B𝑎=9.00
  • C𝑎=0.77
  • D𝑎=64.30
  • E𝑎=1.80

Q18:

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

Q19:

In the following geometric sequence, find the lowest term order, 𝑛, so that the value of the term 𝑎 is less than one,(900,180,36,…).

Q20:

Find the second term of a geometric sequence given 𝑎−𝑎=64, 𝑎−𝑎=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.

  • A3,1,13,⋯, 6
  • B3,1,13,⋯, 5
  • C(3,9,27,…), 5
  • D(3,9,27,…), 6
  • E(3,−9,27,…), 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,⋯.

  • A94
  • B94
  • C94
  • D94

Q24:

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

  • A𝑝=3, 𝑞=5
  • B𝑝=−2, 𝑞=17
  • C𝑝=−7, 𝑞=57
  • D𝑝=5, 𝑞=−37
  • E𝑝=5, 𝑞=37

Q25:

Find the order of the first term whose value is greater than 500 in the geometric sequence (24,48,96,…).

  • A𝑛=6, 𝑎=768
  • B𝑛=5, 𝑎=768
  • C𝑛=5, 𝑎=1,536
  • D𝑛=6, 𝑎=1,536
  • E𝑛=7, 𝑎=1,536

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