Lesson Worksheet: Sum of a Finite Geometric Sequence Mathematics • 10th Grade

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

Q1:

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.

  • A𝑟𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟
  • B𝑟𝑆=𝑎𝑟+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟
  • C𝑟𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟
  • D𝑟𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟
  • E𝑟𝑆=𝑎𝑟+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟

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 not appear on the right-hand side of both equations.

  • A𝑎𝑟, 𝑎𝑟
  • B𝑎, 𝑎𝑟
  • C𝑎, 𝑎𝑟
  • D𝑎, 𝑎𝑟
  • E𝑎𝑟, 𝑎𝑟

Now, consider the subtraction 𝑆𝑟𝑆=𝑎+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟𝑎𝑟+𝑎𝑟+𝑎𝑟+𝑎𝑟++𝑎𝑟.

Use the answer to the previous part to simplify the subtraction 𝑆𝑟𝑆.

  • A𝑆𝑟𝑆=𝑎𝑟𝑎𝑟
  • B𝑆𝑟𝑆=𝑎𝑎𝑟
  • C𝑆𝑟𝑆=𝑎𝑟𝑎𝑟
  • D𝑆𝑟𝑆=𝑎𝑎𝑟
  • E𝑆𝑟𝑆=𝑎𝑎𝑟

Factor both sides of the equation.

  • A𝑆(1𝑟)=𝑎𝑟(1𝑟)
  • B𝑆(1𝑟)=𝑎1𝑟
  • C𝑆(1𝑟)=𝑎𝑟𝑟
  • D𝑆(1𝑟)=𝑎1𝑟
  • E𝑆(1𝑟)=𝑎(1𝑟)

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

  • A𝑆=𝑎𝑟(1𝑟)1𝑟
  • B𝑆=𝑎1𝑟1𝑟
  • C𝑆=𝑎1𝑟1𝑟
  • D𝑆=𝑎𝑟𝑟1𝑟
  • E𝑆=𝑎(1𝑟)1𝑟

Q2:

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.

  • A3692
  • B410
  • C5332
  • D8612

Q3:

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

Q4:

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

Q5:

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

  • A1,3765
  • B1685
  • C3445
  • D1,3445
  • E1725

Q6:

Find the number of terms in the geometric sequence given the first term is 21, the last term is 1516, and the sum of all the terms is 401116.

Q7:

Find the geometric sequence given the sum of all the terms is 3,339, the last term is 1,696, and the common ratio is 2.

  • A6332,6316,638,
  • B153,253,453,
  • C53,106,212,
  • D53,532,534,

Q8:

Fill in the blank: The number of terms of a geometric sequence whose first term is 729, last term is 1, and sum of all terms is 1,093 is .

Q9:

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.

  • A𝑎=1289,649,329,, 𝑆=28
  • B𝑎=136,118,19,, 𝑆=3136
  • C𝑎=2569,1289,649,, 𝑆=56
  • D𝑎=136,118,19,, 𝑆=74
  • E𝑎=1289,649,329,, 𝑆=2489

Q10:

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.

  • A𝑎=913,313,113,, 𝑆=4039
  • B𝑎=113,313,913,, 𝑆=12113
  • C𝑎=113,313,913,, 𝑆=4013
  • D𝑎=913,313,113,, 𝑆=121117
  • E𝑎=2713,8113,24313,, 𝑆=3,26713

Practice Means Progress

Download the Nagwa Practice app to access 96 additional questions for this lesson!

scan me!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.