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=π‘Ž+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘Ÿ.

Multiply the expression for π‘†οŠ by π‘Ÿ, the common ratio.

  • Aπ‘Ÿπ‘†=π‘Ž+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘ŸοŠοŠ¨οŠ©οŠ
  • Bπ‘Ÿπ‘†=π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘ŸοŠοŠ¨οŠ©οŠͺ
  • Cπ‘Ÿπ‘†=π‘Ž+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘ŸοŠοŠ¨οŠ©οŠοŠ±οŠ§
  • Dπ‘Ÿπ‘†=π‘Ž+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘ŸοŠοŠ¨οŠ©οŠͺ
  • Eπ‘Ÿπ‘†=π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘ŸοŠοŠ¨οŠ©οŠͺ

So, we have 𝑆=π‘Ž+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘ŸοŠοŠ¨οŠ©οŠοŠ±οŠ§ and π‘Ÿπ‘†=π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘Ÿ.οŠͺ

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.

  • Aπ‘Žπ‘ŸοŠοŠ±οŠ§, π‘Žπ‘ŸοŠ
  • Bπ‘Ž, π‘Žπ‘ŸοŠ
  • Cπ‘Ž, π‘Žπ‘ŸοŠοŠ±οŠ§
  • Dπ‘Ž, π‘Žπ‘ŸοŠͺ
  • Eπ‘Žπ‘ŸοŠͺ, π‘Žπ‘ŸοŠοŠ±οŠ§

Now, consider the subtraction π‘†βˆ’π‘Ÿπ‘†=ο€Ήπ‘Ž+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘Ÿο…βˆ’ο€Ήπ‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+π‘Žπ‘Ÿ+β‹―+π‘Žπ‘Ÿο….οŠͺ

Use the answer to the previous part to simplify the subtraction π‘†βˆ’π‘Ÿπ‘†οŠοŠ.

  • Aπ‘†βˆ’π‘Ÿπ‘†=π‘Žπ‘Ÿβˆ’π‘Žπ‘ŸοŠοŠοŠοŠ±οŠ§οŠͺ
  • Bπ‘†βˆ’π‘Ÿπ‘†=π‘Žβˆ’π‘Žπ‘ŸοŠοŠοŠͺ
  • Cπ‘†βˆ’π‘Ÿπ‘†=π‘Žπ‘Ÿβˆ’π‘Žπ‘ŸοŠοŠοŠοŠ±οŠ§οŠ
  • Dπ‘†βˆ’π‘Ÿπ‘†=π‘Žβˆ’π‘Žπ‘ŸοŠοŠοŠοŠ±οŠ§
  • Eπ‘†βˆ’π‘Ÿπ‘†=π‘Žβˆ’π‘Žπ‘ŸοŠοŠοŠ

Factor both sides of the equation.

  • A𝑆(1βˆ’π‘Ÿ)=π‘Žπ‘Ÿ(1βˆ’π‘Ÿ)
  • B𝑆(1βˆ’π‘Ÿ)=π‘Žο€Ή1βˆ’π‘Ÿο…οŠοŠοŠ±οŠ§
  • C𝑆(1βˆ’π‘Ÿ)=π‘Žο€Ήπ‘Ÿβˆ’π‘Ÿο…οŠοŠοŠ±οŠ§οŠͺ
  • D𝑆(1βˆ’π‘Ÿ)=π‘Žο€Ή1βˆ’π‘Ÿο…οŠοŠͺ
  • E𝑆(1βˆ’π‘Ÿ)=π‘Ž(1βˆ’π‘Ÿ)

Rearrange the equation to make π‘†οŠ the subject of the formula.

  • A𝑆=π‘Žπ‘Ÿ(1βˆ’π‘Ÿ)1βˆ’π‘ŸοŠοŠοŠ±οŠ§οŠ
  • B𝑆=π‘Žο€Ή1βˆ’π‘Ÿο…1βˆ’π‘ŸοŠοŠοŠ±οŠ§
  • C𝑆=π‘Žο€Ή1βˆ’π‘Ÿο…1βˆ’π‘ŸοŠοŠͺ
  • D𝑆=π‘Žο€Ήπ‘Ÿβˆ’π‘Ÿο…1βˆ’π‘ŸοŠοŠοŠ±οŠ§οŠͺ
  • E𝑆=π‘Ž(1βˆ’π‘Ÿ)1βˆ’π‘ŸοŠοŠ

Q2:

In a geometric sequence, the first term is π‘Ž and the common ratio is π‘Ÿ.

Find the sum of the first 3 terms of a geometric sequence with π‘Ž=328 and π‘Ÿ=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,βˆ’32,64,…,256).

Q5:

Find the sum of the first 7 terms of a geometric sequence given π‘Ž=βˆ’8π‘ŽοŠ«οŠ¨ and π‘Ž+π‘Ž=βˆ’64οŠͺ.

  • A1,3765
  • Bβˆ’1685
  • C3445
  • D1,3445
  • Eβˆ’1725

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.

  • Aπ‘Ž=1289,649,329,β€¦οŠ, 𝑆=28
  • Bπ‘Ž=136,118,19,β€¦οŠ, 𝑆=3136
  • Cπ‘Ž=2569,1289,649,β€¦οŠ, 𝑆=56
  • Dπ‘Ž=136,118,19,β€¦οŠ, 𝑆=74
  • Eπ‘Ž=1289,649,329,β€¦οŠ, 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.

  • Aπ‘Ž=ο€Ό913,313,113,β‹―οˆοŠ, 𝑆=4039
  • Bπ‘Ž=ο€Ό113,313,913,β‹―οˆοŠ, 𝑆=12113
  • Cπ‘Ž=ο€Ό113,313,913,β‹―οˆοŠ, 𝑆=4013
  • Dπ‘Ž=ο€Ό913,313,113,β‹―οˆοŠ, 𝑆=121117
  • Eπ‘Ž=ο€Ό2713,8113,24313,β‹―οˆοŠ, 𝑆=3,26713

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