Partial sums of geometric series

Learning Objectives Identify a geometric series by recognizing a common ratio. Determine the common ratio in a given geometric series. Calculate the next term in a geometric series using the common ratio. Find the partial sum of the first few terms of a geometric series by adding them. Explain what a 'partial sum' means in the context of a series. Apply the concept of geometric series to simple real-world scenarios. Have you ever seen a pattern grow by multiplying? 📈 Like a chain letter where each person sends it to 3 friends, and they each send it to 3 more? That's a geometric series! In this lesson, you'll learn about special number patterns called geometric series, where each number is found by multiplying the previous one by the same amount. We&#039...

TermDefinitionExample SeriesA list of numbers in a specific pattern, usually written with plus signs between them to show they are to be added together.2 + 4 + 6 + 8 + ... is an arithmetic series. 3 + 6 + 12 + 24 + ... is a geometric series. Geometric SeriesA series where each term after the first is found by multiplying the previous term by a fixed, non-zero number.5, 10, 20, 40, ... (Each term is multiplied by 2 to get the next term). TermEach individual number in a series.In the series 3, 9, 27, 81, ..., the number 3 is the first term, 9 is the second term, and so on. Common RatioThe fixed number by which each term in a geometric series is multiplied to get the next term.In the series 2, 6, 18, 54, ..., the common ratio is 3 (because 2 × 3 = 6, 6 × 3 = 18, etc.). Partial SumThe sum of...

Finding the Common Ratio (r) r = \frac{\text{Any term}}{\text{Previous term}} To find the common ratio (r) of a geometric series, divide any term by the term that comes immediately before it. This ratio tells you what number you multiply by to get from one term to the next. Finding the Next Term \text{Next Term} = \text{Current Term} \times r Once you know the common ratio (r), you can find any term in the series by multiplying the term right before it by r. Calculating a Partial Sum (S_n) S_n = a_1 + a_2 + a_3 + ... + a_n To find the partial sum of 'n' terms (S_n), simply add the first 'n' terms of the geometric series together. For Grade 6, we will list out the terms and add them.