Worksheet: Infinite Geometric Series

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In this worksheet, we will practice finding the sum of infinite geometric series.

Q1:

Can the sum of an infinite geometric series with common ratio 𝑟 be found when | 𝑟 | < 1 ?

  • Ayes
  • Bno

Q2:

Find the sum of the infinite geometric sequence 4 9 , − 2 1 , 9 , … .

  • A − 2 1
  • B − 3 4 3 3
  • C 3 4 3 4
  • D 3 4 3 1 0
  • E 1 4 7 1 0

Q3:

Can the sum of an infinite geometric series be found for any value of 𝑟 satisfying | 𝑟 | ≤ 1 ?

  • Ano
  • Byes

Q4:

Find the possible infinite geometric series where the sum is − 2 1 and the sum of the first and second terms is − 3 5 3 .

  • A − 7 , − 2 1 2 , − 6 3 4 , … or − 3 5 , 1 0 5 2 , − 3 1 5 4 , …
  • B − 1 7 , − 3 1 4 , − 9 2 8 , … or − 1 3 5 , 3 7 0 , − 9 1 4 0 , …
  • C − 1 7 , − 2 2 1 , − 4 6 3 , … or − 1 3 5 , 2 1 0 5 , − 4 3 1 5 , …
  • D − 7 , − 1 4 3 , − 2 8 9 , … or − 3 5 , 7 0 3 , − 1 4 0 9 , …

Q5:

Find the sum of an infinite number of terms of the geometric sequence 8 7 , √ 8 7 , 1 , … .

  • A 8 7 8 8 8 7 + √ 8 7
  • B 8 7 8 6 8 7 − √ 8 7
  • C 8 7 8 8 8 7 − √ 8 7
  • D 8 7 8 6 8 7 + √ 8 7
  • E 8 6 8 7 8 7 + √ 8 7

Q6:

Find the common ratio of an infinite geometric sequence given the sum is 52 and the first term is 14.

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

Q7:

Find the infinite geometric sequence given each term of the sequence is twice the sum of the terms that follow it and the first term is 37.

  • A 3 7 , − 3 7 3 , 3 7 9 , …
  • B 3 7 , 1 1 1 , 3 3 3 , …
  • C 1 1 1 , 3 7 , 3 7 3 , …
  • D 3 7 , 3 7 3 , 3 7 9 , …
  • E 1 1 1 , − 3 7 , 3 7 3 , …

Q8:

Find the sum of an infinite number of terms of a geometric sequence given the first term is 2 6 5 and the fourth term is − 6 5 0 3 4 3 .

  • A − 6 3 7 1 5 0
  • B 1 3 6
  • C 9 1 5
  • D 9 1 3 0
  • E − 6 3 7 2 5

Q9:

Find the sum of the infinite geometric sequence 7 , ( − 7 ) , 7 , … − 1 − 3 − 5 .

  • A − 1 3 3 6
  • B 7 4 8
  • C 1 3 5 0
  • D 7 5 0
  • E 6 4 3 5 0

Q10:

Find the sum of the squares of the terms of an infinite geometric sequence given the first term is 49 and the common ratio is 9 𝑥 .

  • A 2 4 0 1 8 1 𝑥 2
  • B 4 9 1 − 9 𝑥
  • C 4 9 9 𝑥
  • D 2 4 0 1 1 − 8 1 𝑥 2
  • E 2 4 0 1 8 1 𝑥 − 1 2

Q11:

Find the infinite geometric sequence given the first term is 5 5 7 and the sum of the terms is 2 0 1 7 1 .

  • A 5 5 7 , − 5 2 2 8 , 5 9 1 2 , …
  • B 5 5 7 , 2 0 5 7 , 8 0 5 7 , …
  • C 5 5 7 , 3 5 2 2 8 , 2 4 5 9 1 2 , …
  • D 5 5 7 , 5 2 2 8 , 5 9 1 2 , …

Q12:

Find the two geometric sequences given 𝑇 + 𝑇 = − 3 5 1 4 and 𝑇 + 𝑇 = − 3 0 2 3 . Then find the sum of an infinite number of terms for the one sequence which can be summed up to infinity.

  • A ( − 2 7 , − 1 8 , − 1 2 , … ) , − 2 7 , − 8 1 2 , − 2 4 3 4 , ⋯ , − 5 4
  • B ( − 2 7 , − 1 8 , − 1 2 , … ) , ( − 8 , − 1 2 , − 1 8 , … ) , 16
  • C ( − 2 7 , − 1 8 , − 1 2 , … ) , − 2 7 , − 8 1 2 , − 2 4 3 4 , ⋯ , − 8 1 5
  • D ( − 2 7 , − 1 8 , − 1 2 , … ) , ( − 8 , − 1 2 , − 1 8 , … ) , − 8 1
  • E ( − 8 , − 1 2 , − 1 8 , … ) , − 2 7 , − 8 1 2 , − 2 4 3 4 , ⋯ , − 1 6 5

Q13:

Find all the possible values of the tenth term of a geometric sequence, given the second term is − 1 2 and the sum of the first three terms is 3 4 .

  • A 1 2 0 4 8 or 512
  • B256 or 1 1 0 2 4
  • C − 1 4 0 9 6 or − 1 0 2 4
  • D − 1 2 8 or − 1 5 1 2

Q14:

Find a geometric sequence given the second term equals 3 2 0 9 and the sum of the first three terms equals − 1 6 0 3 .

  • A 1 6 0 9 , 3 2 0 9 , 6 4 0 9 , ⋯ or − 6 4 0 9 , 3 2 0 9 , − 1 6 0 9 , ⋯
  • B − 1 6 0 9 , 3 2 0 9 , − 6 4 0 9 , ⋯ or 6 4 0 9 , 3 2 0 9 , 1 6 0 9 , ⋯
  • C 1 6 0 9 , 3 2 0 9 , 6 4 0 9 , ⋯ or 6 4 0 9 , 3 2 0 9 , 1 6 0 9 , ⋯
  • D − 1 6 0 9 , 3 2 0 9 , − 6 4 0 9 , ⋯ or − 6 4 0 9 , 3 2 0 9 , − 1 6 0 9 , ⋯

Q15:

Find the infinite geometric sequence given each term of the sequence is half the sum of the terms that follow it and the sum of the second and fourth terms equals 4 3 2 5 7 .

  • A 4 1 6 5 7 , 2 0 8 5 7 , 1 0 4 5 7 , ⋯
  • B 1 3 5 1 9 , 9 0 1 9 , 6 0 1 9 , ⋯
  • C 8 3 2 5 7 , 4 1 6 5 7 , 2 0 8 5 7 , ⋯
  • D 9 0 1 9 , 6 0 1 9 , 4 0 1 9 , ⋯

Q16:

Find the sum of the infinite geometric sequence 1 3 , 1 3 , 1 3 , … − 1 − 2 − 3 .

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

Q17:

Find the sum of an infinite number of terms of the geometric sequence 8 , 2 √ 2 , 1 , … .

  • A 1 6 9 4 + √ 2
  • B 1 6 7 4 − √ 2
  • C 1 6 9 4 − √ 2
  • D 1 6 7 4 + √ 2
  • E 7 1 6 4 + √ 2

Q18:

Find the common ratio of an infinite geometric sequence given the sum is 43 and the first term is 20.

  • A 6 3 4 3
  • B − 1 9 4 3
  • C 4 3 2 0
  • D 2 3 4 3

Q19:

Find the possible infinite geometric series where the sum is 33 and the sum of the first and second terms is 3 8 5 1 2 .

  • A 5 5 2 , 1 6 5 , 9 9 0 , … or 7 7 2 , − 2 3 1 , 1 3 8 6 , …
  • B 2 5 5 , 1 2 5 5 , 7 2 5 5 , … or 2 7 7 , − 1 2 7 7 , 7 2 7 7 , …
  • C 2 5 5 , 1 1 6 5 , 1 9 9 0 , … or 2 7 7 , − 1 2 3 1 , 1 1 3 8 6 , …
  • D 5 5 2 , 5 5 1 2 , 5 5 7 2 , … or 7 7 2 , − 7 7 1 2 , 7 7 7 2 , …

Q20:

Find the possible infinite geometric series where the sum is 16 and the sum of the first and second terms is 8 0 9 .

  • A 1 6 3 , 8 , 1 2 , … or 8 0 3 , − 4 0 , 6 0 , …
  • B 3 1 6 , 9 3 2 , 2 7 6 4 , … or 3 8 0 , − 9 1 6 0 , 2 7 3 2 0 , …
  • C 3 1 6 , 1 8 , 1 1 2 , … or 3 8 0 , − 1 4 0 , 1 6 0 , …
  • D 1 6 3 , 3 2 9 , 6 4 2 7 , … or 8 0 3 , − 1 6 0 9 , 3 2 0 2 7 , …

Q21:

Find the sum of an infinite number of terms of a geometric sequence given the third term is 4 9 and the sixth term is 3 2 2 4 3 .

  • A 9 2
  • B − 2
  • C 3 5
  • D3
  • E 9 1 0

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