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 ๐‘Ž=โˆ’3,616๏Šฌ and the common ratio is โˆ’2.

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

Q2:

Find the value of ๐‘Ž๏Šจ in a geometric sequence given 6๐‘Ž+๐‘Ž=16๐‘Ž,๐‘Ž=62๏Šจ๏Šฉ๏Šง๏Šง๏Šฆ and all terms are positive.

  • A3164
  • Bโˆ’3132
  • C314
  • D31256
  • E31128

Q3:

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

  • Aโˆ’2, 6, โˆ’18
  • B16, โˆ’118, 154
  • Cโˆ’12, 16, โˆ’118
  • D6, โˆ’18, 54
  • E6, โˆ’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 โˆ’7 and the sum of their squares is 21.

  • Aโˆ’14,โˆ’12,โˆ’1
  • Bโˆ’4,โˆ’2,โˆ’1
  • Cโˆ’4,โˆ’8,โˆ’16
  • Dโˆ’14,โˆ’18,โˆ’116

Q6:

Find the order of the term 1639 in the geometric sequence 1156,โˆ’178,139,โ€ฆ.

Q7:

Find the second and third terms of a geometric sequence given ๐‘Ž=69,๐‘Ž=4,416๏Šง๏Šญ, and all terms are positive.

  • A๐‘Ž=692๏Šจ, ๐‘Ž=694๏Šฉ
  • B๐‘Ž=138๏Šจ, ๐‘Ž=276๏Šฉ
  • C๐‘Ž=694๏Šจ, ๐‘Ž=698๏Šฉ
  • D๐‘Ž=138๏Šจ, ๐‘Ž=552๏Šฉ
  • E๐‘Ž=276๏Šจ, ๐‘Ž=552๏Šฉ

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 1โˆถ16. How many terms lie between 38 and 4,864 in the sequence?

Q9:

In a geometric sequence and for a certain ๐‘›, we have ๐‘Ž=6๏Š, ๐‘Ž=3๏Š๏Šฐ๏Šง, and ๐‘Ž=38๏Šจ๏Š๏Šฑ๏Šง. Find the first 3 terms of the sequence and what ๐‘› is.

  • A96,48,24, and ๐‘›=4
  • B192,96,48, and ๐‘›=5
  • C38,34,32, and ๐‘›=5
  • D316,38,34, and ๐‘›=5
  • E96,48,24, and ๐‘›=5

Q10:

Find, in terms of ๐‘›, the general term of the sequence (1,2,4,8,โ€ฆ) where ๐‘›โ‰ฅ1.

  • A(2)๏Š๏Šฑ๏Šง
  • B2๐‘›โˆ’1
  • C2๏Š
  • D2โˆ’1๏Š

Q11:

Determine ๐‘ and ๐‘ž that make the following a geometric sequence, starting at ๐‘Ž๏Šง, of the form ๐‘Ž=๐‘๐‘ž๏Š๏Š๏Šฑ๏Šง: 2,6,18,54,162,โ€ฆ

  • A๐‘=4, ๐‘ž=3
  • B๐‘=3, ๐‘ž=2
  • C๐‘=2, ๐‘ž=2
  • D๐‘=2, ๐‘ž=3
  • E๐‘=3, ๐‘ž=3

Q12:

Find the value of ๐‘› given the ๐‘Ž๏Š of the geometric sequence (96,24,6,โ€ฆ) is three times greater than the ๐‘Ž๏Š of the geometric sequence (4,2,1,โ€ฆ).

Q13:

The sum of the first and second terms of a geometric sequence is 54 and ๐‘Ž+๐‘Ž=216๏Šฉ๏Šช. Find the possible values of ๐‘Ž๏Šฉ.

  • A92 or โˆ’272
  • B144 or 432
  • C72 or โˆ’216
  • D9 or 27

Q14:

Find, in terms of ๐‘›, the general term of the sequence (1,3,9,27,81,โ€ฆ).

  • A3๏Š๏Šฑ๏Šง
  • B3๐‘›
  • C3๏Š
  • D3(๐‘›โˆ’1)

Q15:

Find the order of the term whose value is 4,374 in the geometric sequence ๐‘Ž=23(3)๏Š๏Š.

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,โ€ฆ).

  • A๐‘›=6, ๐‘Ž=23,125๏Šฌ
  • B๐‘›=4, ๐‘Ž=2625๏Šช
  • C๐‘›=4, ๐‘Ž=2125๏Šช
  • D๐‘›=5, ๐‘Ž=2625๏Šซ
  • E๐‘›=5, ๐‘Ž=23,125๏Šซ

Q17:

Suppose that ๐‘Ž=5๏Šซ and ๐‘Ž=3๏Šญ belong to a geometric sequence with a positive ๐‘Ž๏Šง. Using four decimal places for your common ratio, determine this first term to two decimal places.

  • A๐‘Ž=13.89๏Šง
  • B๐‘Ž=9.00๏Šง
  • C๐‘Ž=0.77๏Šง
  • D๐‘Ž=64.30๏Šง
  • E๐‘Ž=1.80๏Šง

Q18:

Find the number of terms of the geometric sequence ๏€ผ112,56,28,โ€ฆ,74๏ˆ.

Q19:

In the following geometric sequence, find the lowest term order, ๐‘›, so that the value of the term ๐‘Ž๏Š is less than one,(900,180,36,โ€ฆ).

Q20:

Find the second term of a geometric sequence given ๐‘Žโˆ’๐‘Ž=64๏Šซ๏Šช, ๐‘Žโˆ’๐‘Ž=48๏Šช๏Šจ 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๏€ผ3,1,13,โ‹ฏ๏ˆ, 6
  • B๏€ผ3,1,13,โ‹ฏ๏ˆ, 5
  • C(3,9,27,โ€ฆ), 5
  • D(3,9,27,โ€ฆ), 6
  • E(3,โˆ’9,27,โ€ฆ), 6

Q22:

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

Q23:

Find, in terms of ๐‘›, the general term of the sequence ๏€ผ14,916,8164,729256,โ‹ฏ๏ˆ.

  • A94๏Š๏Šฑ๏Šง๏Š๏Šฐ๏Šง
  • B94๏Š๏Šฑ๏Šง๏Š
  • C94๏Š๏Š๏Šฑ๏Šง
  • D94๏Š๏Šฐ๏Šง๏Š

Q24:

Determine ๐‘ and ๐‘ž that make 5,โˆ’157,4549,โˆ’135343,4052,401,โ€ฆ a geometric sequence, starting at ๐‘Ž๏Šง, of the form ๐‘Ž=๐‘๐‘ž๏Š๏Š๏Šฑ๏Šง.

  • A๐‘=3, ๐‘ž=5
  • B๐‘=โˆ’2, ๐‘ž=17
  • C๐‘=โˆ’7, ๐‘ž=57
  • D๐‘=5, ๐‘ž=โˆ’37
  • E๐‘=5, ๐‘ž=37

Q25:

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

  • A๐‘›=6, ๐‘Ž=768๏Šฌ
  • B๐‘›=5, ๐‘Ž=768๏Šซ
  • C๐‘›=5, ๐‘Ž=1,536๏Šซ
  • D๐‘›=6, ๐‘Ž=1,536๏Šฌ
  • E๐‘›=7, ๐‘Ž=1,536๏Šญ

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