Worksheet: Sum of a Finite Geometric Sequence

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In this worksheet, we will practice calculating the sum of the terms in a geometric sequence with a definite number of terms.

Q1:

The sum of the terms of a sequence is called a series.

A geometric series is the sum of a geometric sequence; a geometric series with 𝑛 terms can be written as 𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟+⋯+𝑎𝑟,() where 𝑎 is the first term and 𝑟 is the common ratio (the number you multiply one term by to get the next term in the sequence, 𝑟≠1).

Find the sum of the first 6 terms of a geometric series with 𝑎=24 and 𝑟=12.

  • A 8 1 2
  • B 4 7 1 4
  • C 2 3 1 4
  • D204
  • E 1 1 5 8

Q2:

A geometric series has a first term of 3 and a common ratio of 5. Find the sum of the first 6 terms.

Q3:

Find, to two decimal places, the sum of the following geometric series: 20+20(1.01)+20(1.01)+⋯+20(1.01).

Q4:

We can find a formula for the sum of a geometric series. Consider the series 𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟+⋯+𝑎𝑟.

Multiply the expression for 𝑆 by 𝑟, the common ratio.

  • A 𝑟 𝑆 = 𝑎 + 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + ⋯ + 𝑎 𝑟
  • B 𝑟 𝑆 = 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + ⋯ + 𝑎 𝑟
  • C 𝑟 𝑆 = 𝑎 + 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + ⋯ + 𝑎 𝑟
  • D 𝑟 𝑆 = 𝑎 + 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + ⋯ + 𝑎 𝑟
  • E 𝑟 𝑆 = 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + 𝑎 𝑟 + ⋯ + 𝑎 𝑟

So, we have 𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟+⋯+𝑎𝑟 and 𝑟𝑆=𝑎𝑟+𝑎𝑟+𝑎𝑟+𝑎𝑟+⋯+𝑎𝑟.

The right-hand sides of the equations are very similar. Identify the terms that do \textbf{not} appear on the right-hand side of both equations.

  • A 𝑎 𝑟 , 𝑎 𝑟
  • B 𝑎 , 𝑎 𝑟
  • C 𝑎 , 𝑎 𝑟
  • D 𝑎 , 𝑎 𝑟
  • E 𝑎 𝑟 , 𝑎 𝑟

Now, consider the subtraction 𝑆−𝑟𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟+⋯+𝑎𝑟−𝑎𝑟+𝑎𝑟+𝑎𝑟+𝑎𝑟+⋯+𝑎𝑟.

Use the answer to the previous part to simplify the subtraction 𝑆−𝑟𝑆.

  • A 𝑆 − 𝑟 𝑆 = 𝑎 𝑟 − 𝑎 𝑟
  • B 𝑆 − 𝑟 𝑆 = 𝑎 − 𝑎 𝑟
  • C 𝑆 − 𝑟 𝑆 = 𝑎 𝑟 − 𝑎 𝑟
  • D 𝑆 − 𝑟 𝑆 = 𝑎 − 𝑎 𝑟
  • E 𝑆 − 𝑟 𝑆 = 𝑎 − 𝑎 𝑟

Factor both sides of the equation.

  • A 𝑆 ( 1 − 𝑟 ) = 𝑎 𝑟 ( 1 − 𝑟 )
  • B 𝑆 ( 1 − 𝑟 ) = 𝑎 1 − 𝑟
  • C 𝑆 ( 1 − 𝑟 ) = 𝑎 𝑟 − 𝑟
  • D 𝑆 ( 1 − 𝑟 ) = 𝑎 1 − 𝑟
  • E 𝑆 ( 1 − 𝑟 ) = 𝑎 ( 1 − 𝑟 )

Rearrange the equation to make 𝑆 the subject of the formula.

  • A 𝑆 = 𝑎 𝑟 ( 1 − 𝑟 ) 1 − 𝑟
  • B 𝑆 = 𝑎 1 − 𝑟 1 − 𝑟
  • C 𝑆 = 𝑎 1 − 𝑟 1 − 𝑟
  • D 𝑆 = 𝑎 𝑟 − 𝑟 1 − 𝑟
  • E 𝑆 = 𝑎 ( 1 − 𝑟 ) 1 − 𝑟

Q5:

There are two geometric series with a first term of 3 and whose first 3 terms have a sum of 21.

What are their common ratios?

  • A3, 7
  • B − 2 , − 3
  • C − 2 , 3
  • D2, −3
  • E − 3 , − 7

Write an expression for the sum of the first 𝑛 terms of the sequence with first term 3 and common ratio 2.

  • A 3 2 − 1
  • B 3 2
  • C 2 ( 1 − 3 )
  • D 2 3
  • E 2 3 − 1

Q6:

In a geometric sequence, the first term is 𝑎 and the common ratio is 𝑟.

Find the sum of the first 3 terms of a geometric sequence with 𝑎=328 and 𝑟=14.

  • A 3 6 9 2
  • B410
  • C 5 3 3 2
  • D 8 6 1 2

Q7:

Find the geometric sequence given the first term is 324, the last term is 4, and the sum of all the terms is 484.

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

Q8:

Find the fourth term of the geometric sequence given by 𝑆=1,024−2 where 𝑆 is the sum of the first 𝑛 terms.

Q9:

Find the first three positive terms of a geometric sequence given their sum is 14 and the sum of their multiplicative inverses is 6332.

  • A 3 2 , 6, and 24
  • B 3 2 , 3 8 , and 332
  • C 2 3 , 8 3 , and 323
  • D 2 3 , 1 6 , and 124

Q10:

Find, in terms of 𝑙 and 𝑚, the common ratio of the geometric sequence given the sum of the first ten terms is 24𝑙 and the sum of the next ten terms is 91𝑚.

  • A 2 4 𝑙 9 1 𝑚
  • B 2 4 𝑙 9 1 𝑚
  • C 9 1 𝑚 2 4 𝑙
  • D 2 4 𝑚 9 1 𝑙
  • E 9 1 𝑚 2 4 𝑙

Q11:

Find the geometric sequence and the sum of the first six terms given the second term is four times the fourth one, the sum of the fourth and seventh terms is 2 and all terms are positive.

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

Q12:

Find the sum of the first 20 terms of the geometric sequence 1,1.07,1.07,1.07,… giving the answer to two decimal places.

Q13:

Find the sum of the geometric series 322−122+166−⋯−11,782.

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

Q14:

Find the sum of the geometric series 176+88+44+⋯+11.

Q15:

Find the geometric sequence and the sum of the first six terms given that the sum of the second and fourth terms is −68 and the sum of the third and fifth terms is −272.

  • A 𝑎 = ( − 2 5 6 , − 6 4 , − 1 6 , … ) , 𝑆 = − 3 4 1
  • B 𝑎 = − 1 4 , − 1 , − 4 , … , 𝑆 = − 1 , 3 6 5 4
  • C 𝑎 = ( − 1 , − 4 , − 1 6 , … ) , 𝑆 = − 3 4 1
  • D 𝑎 = ( − 1 , − 4 , − 1 6 , … ) , 𝑆 = − 1 , 3 6 5
  • E 𝑎 = ( − 2 5 6 , − 6 4 , − 1 6 , … ) , 𝑆 = − 1 , 3 6 5 4

Q16:

Find the sum of the first 7 terms of a geometric sequence given 𝑎=−8𝑎 and 𝑎+𝑎=−64.

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

Q17:

Find the number of terms of the geometric sequence 𝑎=96×2 for which the sum is equal to 93.

Q18:

Find the least number of terms of the geometric sequence given the first term is 1 and the fourth term is 125, where the sum of the terms is 7,143.

Q19:

In a geometric sequence, the first term is 𝑎, the common ratio is 𝑟, and the last term is 𝑙.

Find the sum of the geometric sequence with 𝑎=1,408, 𝑟=12, and 𝑙=88.

Q20:

Find the geometric sequence and the sum of the first seven terms given 15𝑎−6𝑎=−9𝑎, 𝑎=9 and all terms are positive.

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

Q21:

Find the sum of the first 6 terms of the geometric series 12+14+18+116+⋯.

  • A 3 1 1 2
  • B 3 1 3 2
  • C 1 5 1 2
  • D 6 3 6 4

Q22:

Find the geometric sequence and the sum of the first five terms given the sum of the first three terms is 1 and the sum of the next three terms is 27.

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

Q23:

Find the number of terms of the geometric sequence (23,69,207,…) for which the sum is equal to 2,783.

Q24:

Find the sum of the first 6 terms of the geometric sequence (405,135,45,…).

  • A 1 6 0 3
  • B 1 , 8 2 0 3
  • C 1 , 8 2 5 6
  • D605

Q25:

Find the sum of the geometric sequence (16,−32,64,…,256).

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