Worksheet: Infinite Geometric Series

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In this worksheet, we will practice determining whether a geometric series is convergent and finding its limit.

Q1:

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

  • Ano
  • Byes

Q2:

Find the sum of the infinite geometric sequence 49,−21,9,….

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

Q3:

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

  • Ayes
  • Bno

Q4:

Find the possible infinite geometric series where the sum is −21 and the sum of the first and second terms is −353.

  • A − 1 7 , − 2 2 1 , − 4 6 3 , … or −135,2105,−4315,…
  • B − 7 , − 1 4 3 , − 2 8 9 , … or −35,703,−1409,…
  • C − 1 7 , − 3 1 4 , − 9 2 8 , … or −135,370,−9140,…
  • D − 7 , − 2 1 2 , − 6 3 4 , … or −35,1052,−3154,…

Q5:

Find the sum of an infinite number of terms of the geometric sequence 87,√87,1,….

  • A 8 6 8 7 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 7 8 8 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 2 6 7
  • B − 1 4
  • C 1 9 2 6
  • D 3 3 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 1 1 1 , − 3 7 , 3 7 3 , …
  • C 3 7 , 1 1 1 , 3 3 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 265 and the fourth term is −650343.

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

Q9:

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

  • A 6 4 3 5 0
  • B 1 3 5 0
  • C − 1 3 3 6
  • D 7 4 8
  • E 7 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 9𝑥.

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

Q11:

Find the infinite geometric sequence given the first term is 557 and the sum of the terms is 20171.

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

Q12:

Find the two geometric sequences given 𝑎+𝑎=−35 and 𝑎+𝑎=−30. Then find the sum of an infinite number of terms for the one sequence which can be summed up to infinity.

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

Q13:

Find a geometric sequence given the second term equals 3209 and the sum of the first three terms equals −1603.

  • A − 1 6 0 9 , 3 2 0 9 , − 6 4 0 9 , ⋯ or −6409,3209,−1609,⋯
  • B 1 6 0 9 , 3 2 0 9 , 6 4 0 9 , ⋯ or 6409,3209,1609,⋯
  • C 1 6 0 9 , 3 2 0 9 , 6 4 0 9 , ⋯ or −6409,3209,−1609,⋯
  • D − 1 6 0 9 , 3 2 0 9 , − 6 4 0 9 , ⋯ or 6409,3209,1609,⋯

Q14:

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 43257.

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

Q15:

Find the sum of the infinite geometric sequence 13,13,13,….

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

Q16:

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

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

Q17:

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

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

Q18:

Find the possible infinite geometric series where the sum is 33 and the sum of the first and second terms is 38512.

  • A 2 5 5 , 1 1 6 5 , 1 9 9 0 , … or 277,−1231,11,386,…
  • B 5 5 2 , 5 5 1 2 , 5 5 7 2 , … or 772,−7712,7772,…
  • C 2 5 5 , 1 2 5 5 , 7 2 5 5 , … or 277,−1277,7277,…
  • D 5 5 2 , 1 6 5 , 9 9 0 , … or 772,−231,1,386,…

Q19:

Find the possible infinite geometric series where the sum is 16 and the sum of the first and second terms is 809.

  • A 3 1 6 , 1 8 , 1 1 2 , … or 380,−140,160,…
  • B 1 6 3 , 3 2 9 , 6 4 2 7 , … or 803,−1609,32027,…
  • C 3 1 6 , 9 3 2 , 2 7 6 4 , … or 380,−9160,27320,…
  • D 1 6 3 , 8 , 1 2 , … or 803,−40,60,…

Q20:

Find the sum of an infinite number of terms of a geometric sequence given the third term is 49 and the sixth term is 32243.

  • A 9 1 0
  • B 3 5
  • C3
  • D 9 2
  • E − 2

Q21:

Is the infinite geometric series 10+6+185+5425+162125+⋯ convergent?

  • Ayes
  • Bno

Q22:

Is the series 884+8849+88481+⋯ convergent or divergent?

  • Aconvergent
  • Bdivergent

Q23:

If possible, find the sum of the series ∞(−2)3.

  • A − 2 3
  • B 3 5
  • C3
  • D 1 5
  • EThe series diverges.

Q24:

Express the series −12+14−18+116−⋯ in sigma notation .

  • A − 1 2
  • B ∞ 1 2
  • C ∞ − 1 2
  • D 1 2

Q25:

Expand ∞−15.

  • A ( 1 5 , 6 2 5 − 3 , 1 2 5 + 6 2 5 − 1 2 5 + ⋯ )
  • B ( 1 5 , 6 2 5 − 3 , 1 2 5 + 6 2 5 − 1 2 5 )
  • C ( 1 5 , 6 2 5 , − 3 , 1 2 5 , 6 2 5 , − 1 2 5 )
  • D ( 1 5 , 6 2 5 , − 3 , 1 2 5 , 6 2 5 , − 1 2 5 , ⋯ )

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