Lesson: The Sum of an Infinite Geometric Series

In this lesson, we will learn how to find the sum of an infinite geometric series.

Sample Question Videos

  • 01:50

Worksheet: The Sum of an Infinite Geometric Series • 25 Questions • 1 Video

Q1:

Find the sum of the geometric series 1 3 2 + 1 3 4 + 1 3 8 + .

Q2:

Find the sum of the five consecutive terms that start from the third term in the geometric sequence 1 5 1 1 , 3 0 1 1 , 6 0 1 1 , .

Q3:

For which of the following series can the sum to infinity be calculated?

Q4:

Which of the following geometric sequences can be summed up to infinity?

Q5:

Find how many terms should be taken from the geometric sequence 4 1 , 8 2 , 1 6 4 , starting from the first term to make the sum greater than 4 456.

Q6:

Find the sum of an infinite number of terms of a geometric sequence starting from the third term given the third term is 1 3 7 2 and the sixth term is 1 3 5 7 6 .

Q7:

Find how many terms should be taken from the geometric sequence ( 0 . 3 , 0 . 9 , 2 . 7 , ) starting from the first term to make the sum 327.9.

Q8:

The 𝑛 th term of a geometric sequence is 𝑎 , first term is 𝑎 , and the sum of the first 𝑛 terms is 𝑆 .

Find the first three terms and the sum to infinity ( 𝑆 ) of the infinite geometric sequence with 𝑎 𝑎 = 3 3 and 𝑆 = 1 3 2 .

Q9:

Fares started a job with an annual salary of 6 892 LE with a constant annual raise of 207 LE. Maged started with the same annual salary with an annual rise of 4 % of the salary of the year before. Who has a higher income after 13 years and what is the difference in incomes giving the answer to the nearest pound?

Q10:

Find the infinite geometric sequence and the sum to infinity given the sum of the first and second terms is 52, the sum of the third and fourth terms is 13, and all terms are positive.

Q11:

Find the sum of an infinite geometric sequence given the first term is 171 and the fourth term is 1 7 1 6 4 .

Q12:

Find the sum of an infinite number of terms of the geometric sequence starting at 𝑇 1 with 𝑛 t h term 𝑇 = 3 × 1 4 𝑛 1 𝑛 .

Q13:

Find how many terms should be taken from the geometric sequence 5 8 4 , 5 8 4 7 , 5 8 4 4 9 , starting from the second term to make the sum 7 3 .

Q14:

Find the two geometric sequences given the sum of the first three terms in each one is 21 and the sum of the squares of the same terms in each one is 819. Then find the sum of an infinite number of terms for the one sequence which can be summed up to infinity.

Q15:

Find the sum of the first five terms of a geometric sequence given the second term is 27, the third term exceeds the first by 72, and all terms are positive.

Q16:

Find the sum of an infinite geometric sequence given 𝑇 = 6 5 7 1 and 𝑇 = 2 𝑇 𝑛 𝑛 + 1 .

Q17:

If possible, find the sum of the series 𝑛 = 1 𝑛 𝑛 𝑛 2 + 4 5 .

Q18:

The figure shows the steps to producing a curve 𝐶 . It starts as the boundary of the unit square in Figure (a). In Figure (b), we remove a square quarter of the area of the square in (a). In Figure (c), we add a square quarter of the area that we removed in (b). In Figure (d), we remove a square quarter of the area of the square we added in (c). If we continue to do this indefinitely, we will get the curve 𝐶 . We let 𝑅 be the region enclosed by 𝐶 . By summing a suitable infinite series, find the area of region 𝑅 . Give your answer as a fraction.

Q19:

If possible, find the sum of the series 𝑛 = 1 𝑛 1 𝑛 ( 2 ) 3 .

Q20:

Consider the series 4 + 𝑦 + 𝑦 + 𝑦 + 𝑦 + 2 3 4 .

Is this series geometric?

The series is convergent for some 𝑦 . What does it converge to in those cases? Give a simplified answer.

Q21:

Consider the series 1 6 0 + 1 6 0 2 + 8 0 + 8 0 2 + 4 0 + 4 0 2 + .

The series is geometric. What is its common ratio?

Is this series convergent? If yes, what is its sum?

Q22:

Find the geometric sequence and the sum to infinity given the sum of the first three terms equals 42, the first term exceeds the second term by 24, and all terms are positive.

Q23:

Find the sum of the geometric series 6 5 5 8 6 5 2 3 2 + 6 5 9 2 8 .

Q24:

Find the sum of the geometric series 2 3 4 8 + 2 3 1 6 + 6 9 1 6 + .

Q25:

Which of the following geometric sequences can be summed up to infinity?

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