Introduction to partial sums

Learning Objectives Identify a sequence of numbers and its individual terms. Define what a partial sum is in the context of a sequence. Calculate the first, second, and third partial sums of a given sequence. Explain the difference between a term in a sequence and a partial sum. Apply the concept of partial sums to simple real-world scenarios. Recognize how partial sums represent cumulative totals. Accurately perform addition to find partial sums. Have you ever kept a running total of your daily steps 🚶‍♀️ or tracked how much money you've saved over a few days? That's exactly what we'll explore today! In this lesson, you'll learn about partial sums, which are like cumulative totals of numbers in an ordered list. Understanding partial sums helps us see...

TermDefinitionExample SequenceAn ordered list of numbers. Each number in the list is called a term.The sequence 2, 4, 6, 8, ... is an ordered list of even numbers. Term (of a sequence)Each individual number in a sequence. We often use 'a' with a small number (index) to show its position, like $a_1$ for the first term.In the sequence 5, 10, 15, 20, ..., the first term ($a_1$) is 5, and the second term ($a_2$) is 10. Partial SumThe sum of a certain number of terms from the beginning of a sequence. It's like a running total.For the sequence 1, 2, 3, 4, ..., the partial sum of the first two terms is $1 + 2 = 3$. First Partial Sum ($S_1$)The sum of only the first term of a sequence. It is equal to the first term itself.For the sequence 7, 14, 21, ..., the first partial sum ($S_1...

Calculating the First Partial Sum $S_1 = a_1$ The first partial sum is simply the value of the first term in the sequence. Calculating the Second Partial Sum $S_2 = a_1 + a_2$ To find the second partial sum, add the first term and the second term of the sequence. Calculating the Nth Partial Sum (Explicit) $S_n = a_1 + a_2 + ... + a_n$ To find the Nth partial sum, add up all the terms from the first term ($a_1$) up to the Nth term ($a_n$). The 'n' tells you how many terms to include in your sum. Calculating the Nth Partial Sum (Recursive) $S_n = S_{n-1} + a_n$ (for $n > 1$) You can also find the Nth partial sum by adding the Nth term ($a_n$) to the previous partial sum ($S_{n-1}$). This shows the cumulative nature of partial sums.