Partial sums of geometric series

Learning Objectives Define a partial sum of a geometric series. Identify the first term (a₁), common ratio (r), and number of terms (n) for a given geometric series. Apply the formula to calculate the partial sum (S_n) of a finite geometric series. Use summation (sigma) notation to represent a geometric series and calculate its sum. Solve for a₁, r, or n given the partial sum and other variables. By the aend of this lesson, students will be able to model and solve real-world problems involving partial sums of geometric series. If a viral video gets 10 views in the first hour and triples its views every hour, how many total views does it have after 12 hours? 🤔 Calculating this one by one would be tedious! This tutorial explores partial sums of geometric series, which are po...

TermDefinitionExample Geometric SequenceA sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.The sequence 2, 6, 18, 54, ... is geometric because each term is 3 times the previous one. Common Ratio (r)The constant factor you multiply by to get from one term to the next in a geometric sequence. It can be found by dividing any term by its preceding term (r = a_k / a_{k-1}).In the sequence 100, 50, 25, ..., the common ratio is r = 50 / 100 = 0.5. Geometric SeriesThe expression formed by adding the terms of a geometric sequence.For the sequence 2, 6, 18, 54, the corresponding geometric series is 2 + 6 + 18 + 54. Partial Sum (S_n)The sum of a specified number of terms from the beginning of a sequence....

Formula for the nth Partial Sum S_n = a_1 \frac{1 - r^n}{1 - r} This is the primary formula used to find the sum of the first 'n' terms of a geometric series. It is valid for any common ratio 'r' not equal to 1. Alternate Formula for the nth Partial Sum S_n = \frac{a_1 - a_n r}{1 - r} This version is useful when you know the first term (a₁), the last term (a_n), and the common ratio (r), but not the number of terms (n). It is also valid for r ≠ 1. Special Case for r = 1 S_n = n \cdot a_1 If the common ratio is 1, the series is just the first term added to itself 'n' times (e.g., 5 + 5 + 5 + 5). The sum is simply the number of terms multiplied by the first term.