Worksheet: Introduction to Geometric Sequences

In this worksheet, we will practice identifying geometric sequences and relating them to their graphical representations.

Q1:

Find the common ratio of the geometric sequence 𝑎 = 1 1 5 6 , 1 5 2 , 3 5 2 , 9 5 2 , 2 7 5 2 .

  • A2
  • B 1 3
  • C8
  • D3

Q2:

Find the next term of the geometric sequence 5 , 5 4 , 5 1 6 , 5 6 4 , .

  • A 1 , 2 8 0
  • B 5 1 , 0 2 4
  • C 5 , 1 2 0
  • D 5 2 5 6

Q3:

Find the next four terms in the geometric sequence 1 1 6 5 , 1 5 5 , 3 5 5 , .

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

Q4:

Find the fifth term of the geometric sequence 1 8 6 , 1 4 3 , 2 4 3 , .

  • A 4 4 3
  • B 1 6 4 3
  • C 1 7 8 6
  • D 8 4 3

Q5:

State whether the following is true or false: The terms of a geometric sequence can be plotted as a set of collinear points.

  • Afalse
  • Btrue

Q6:

Find the next term in the sequence 6 , 3 0 , 1 5 0 , 7 5 0 , .

Q7:

Which of the following is a geometric sequence?

  • A 𝑎 = 𝑛 3 , where 𝑛 2 .
  • B 𝑎 = 3 ( 𝑛 + 3 ) , where 𝑛 1 .
  • C 𝑎 = 𝑛 ( 𝑛 + 2 ) , where 𝑛 1 .
  • D 𝑎 = 5 𝑎 , where 𝑛 2 .

Q8:

State whether the following is true or false: A geometric sequence is decreasing if its common ratio 𝑟 ( 1 , 0 ) .

  • AFalse
  • BTrue

Q9:

Find the common ratio of a geometric sequence given the middle terms are 56 and 168 respectively.

  • A 7 3
  • B 1 3
  • C 3 7
  • D3
  • E112

Q10:

State whether the following is true or false: A geometric sequence is alternating if its common ratio 𝑟 satisfies 𝑟 ] 1 , 0 [ .

  • Atrue
  • Bfalse

Q11:

For an increasing geometric sequence with first term 𝑎 , and common ratio 𝑟 , which of the following could be true?

  • A 𝑎 > 0 , 0 < 𝑟 < 1
  • B 𝑎 > 0 , 1 < 𝑟 < 0
  • C 𝑎 < 0 , 1 < 𝑟 < 0
  • D 𝑎 < 0 , 0 < 𝑟 < 1
  • E 𝑎 < 1 , 1 < 𝑟 < 0

Q12:

Find the infinite geometric sequence given the first term exceeds the second term by 12, the sum of its terms is 48, and all terms are positive.

  • A ( 3 6 , 2 4 , 1 2 , )
  • B ( 2 4 , 3 6 , 5 4 , )
  • C 1 2 4 , 1 1 2 , 1 6 ,
  • D ( 2 4 , 1 2 , 6 , )

Q13:

Find the infinite geometric sequence and the sum given 𝑎 = 8 and 𝑎 = 1 .

  • A 𝑎 = 8 3 , 1 6 3 , 3 2 3 , , 𝑆 = 8 3
  • B 𝑎 = ( 6 4 , 3 2 , 1 6 , ) , 𝑆 = 1 2 8
  • C 𝑎 = ( 1 6 , 3 2 , 6 4 , ) , 𝑆 = 1 6
  • D 𝑎 = ( 3 2 , 1 6 , 8 , ) , 𝑆 = 6 4

Q14:

Find the infinite geometric sequence given each of its terms is six times the sum of the terms that follow it, the second term equals the multiplicative inverse of the fourth term, and all terms are positive. Then find the sum of the first five terms.

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

Q15:

Find the infinite geometric sequence given the sum of the terms is 8 and the sum of the squares to infinity is 32.

  • A 3 2 3 , 3 2 9 , 3 2 2 7 ,
  • B ( 1 6 , 4 8 , 1 4 4 , )
  • C ( 3 2 , 9 6 , 2 8 8 , )
  • D 1 6 3 , 1 6 9 , 1 6 2 7 ,

Q16:

Find the geometric sequence and the sum of the first six terms given the sixth term is 2 464 and the ninth term is 19 712.

  • A 𝑇 = 7 7 , 7 7 2 , 7 7 4 , 𝑛 , 𝑆 = 4 8 5 1 3 2 6
  • B 𝑇 = ( 7 7 , 1 5 4 , 3 0 8 , ) 𝑛 , 𝑆 = 2 3 8 7 6
  • C 𝑇 = 7 7 , 7 7 2 , 7 7 4 , 𝑛 , 𝑆 = 2 3 8 7 1 6 6
  • D 𝑇 = ( 7 7 , 1 5 4 , 3 0 8 , ) 𝑛 , 𝑆 = 4 8 5 1 6
  • E 𝑇 = 1 7 7 , 2 7 7 , 4 7 7 , 𝑛 , 𝑆 = 3 7 6

Q17:

Find the infinite geometric sequence given the sum of its terms is 4 0 and the sum of their cubes is 1 9 2 , 0 0 0 .

  • A ( 2 0 , 1 0 , 5 , )
  • B ( 1 2 0 , 2 4 0 , 4 8 0 , )
  • C ( 4 0 , 8 0 , 1 6 0 , )
  • D ( 6 0 , 3 0 , 1 5 , )
  • E ( 6 0 , 3 0 , 1 5 , ) or ( 1 2 0 , 2 4 0 , 4 8 0 , )

Q18:

Find the geometric sequence that has an infinite number of terms and its sum to infinity, where the sum of the second and third terms is 20 and the sum of the first three terms is 38.

  • A 𝑎 = 2 5 0 3 , 5 0 , 3 0 , , 𝑆 = 6 2 5 1 2
  • B 𝑎 = 1 8 , 3 0 , 5 0 , , 𝑆 = 2 7 4
  • C 𝑎 = 1 8 , 1 2 , 8 , , 𝑆 = 5 4 5
  • D 𝑎 = 1 8 , 1 2 , 8 , , 𝑆 = 5 4
  • E 𝑎 = 2 5 0 3 , 5 0 , 3 0 , , 𝑆 = 2 7

Q19:

Find the geometric sequence given by 𝑆 = 6 , 5 6 1 9 , where 𝑆 is the sum of the first 𝑛 terms.

  • A ( 5 , 8 3 2 , 6 4 8 , 6 , 5 5 2 , )
  • B ( 5 , 8 3 2 , 6 , 4 8 0 , 6 , 5 5 2 , )
  • C ( 5 , 8 3 2 , 6 4 8 , 6 , 4 8 0 , )
  • D ( 5 , 8 3 2 , 6 4 8 , 7 2 , )

Q20:

Find two geometric sequences given the sum of the first and third terms in each one is 180 and the sum of the first three terms in each one is 234. Then find the sum of an infinite number of terms for the one sequence which can be summed up to infinity.

  • A ( 1 6 2 , 5 4 , 1 8 , ) , ( 1 8 , 5 4 , 1 6 2 , ) , 2 4 3 2
  • B ( 1 6 2 , 5 4 , 1 8 , ) , ( 1 8 , 5 4 , 1 6 2 , ) , 9
  • C ( 1 6 2 , 5 4 , 1 8 , ) , ( 1 8 , 5 4 , 1 6 2 , ) , 9 2
  • D ( 1 6 2 , 5 4 , 1 8 , ) , ( 1 8 , 5 4 , 1 6 2 , ) , 243
  • E ( 1 6 2 , 5 4 , 1 8 , ) , ( 1 8 , 5 4 , 1 6 2 , ) , 243

Q21:

Find the two geometric sequences in which the product of the first three terms in each one is 1 , 7 2 8 and the sum of the second, third, and fourth terms in each one is 2 1 . Then, find the sum of an infinite number of terms for the one sequence which can be summed up to infinity.

  • A ( 1 6 , 1 3 , 2 3 , ) , ( 1 1 8 , 1 2 7 , 2 8 1 , ) , 1 2 4
  • B ( 2 4 , 1 2 , 6 , ) , ( 8 , 1 2 , 1 8 , ) , 1 6 5
  • C ( 1 6 , 1 3 , 2 3 , ) , ( 1 1 8 , 1 2 7 , 2 8 1 , ) , 3 4 0
  • D ( 2 4 , 1 2 , 6 , ) , ( 8 , 1 2 , 1 8 , ) , 4 8
  • E ( 2 4 , 1 2 , 6 , ) , ( 1 1 8 , 1 2 7 , 2 8 1 , ) , 4 8

Q22:

Find a positive geometric sequence and the sum of the first eight terms given 𝑎 = 4 and 𝑎 𝑎 = 6 .

  • A 𝑎 = 2 , 1 , 1 2 , , 𝑆 = 2 5 5 6 4
  • B 𝑎 = 2 , 1 , 1 2 , , 𝑆 = 1 7 0
  • C 𝑎 = ( 8 , 4 , 2 , ) , 𝑆 = 2 5 7 4 8
  • D 𝑎 = ( 8 , 4 , 2 , ) , 𝑆 = 2 5 5 1 6

Q23:

Find the geometric sequence given the sum of the first five terms is 30.5 and the sum of the next five terms is 9 7 6 .

  • A 2 2 6 1 , 4 4 6 1 , 8 8 6 1 ,
  • B 6 1 2 2 , 6 1 4 4 , 6 1 8 8 ,
  • C 2 2 6 1 , 1 1 6 1 , 1 1 1 2 2 ,
  • D 6 1 2 2 , 6 1 1 1 , 1 2 2 1 1 ,

Q24:

Find the sequence and the sum of the first five terms of an infinite geometric sequence given the sum of terms is 144 and the first term is greater than the second term by 36.

  • A 𝑎 = ( 7 2 , 1 0 8 , 1 6 2 , ) , 𝑆 = 1 , 8 9 9 2
  • B 𝑎 = ( 7 2 , 3 6 , 1 8 , ) , 𝑆 = 1 3 5
  • C 𝑎 = ( 7 2 , 1 0 8 , 1 6 2 , ) , 𝑆 = 5 8 5
  • D 𝑎 = ( 7 2 , 3 6 , 1 8 , ) , 𝑆 = 2 7 9 2
  • E 𝑎 = 4 8 , 3 2 , 6 4 3 , , 𝑆 = 3 , 3 7 6 2 7

Q25:

Find the infinite geometric sequence given the first term of the sequence is the sum of the terms that follow it and the sum of the first and second terms is 1 7 5 .

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

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