Worksheet: Finding the nth Term of a Geometric Sequence

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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 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.

  • 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 𝑛≥1.

  • 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 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.

  • 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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