Introduction to partial sums

Learning Objectives Define a partial sum and distinguish it from a term in a sequence. Calculate the nth partial sum (S_n) for a given sequence by direct addition. Represent a partial sum using formal summation (Sigma) notation. Generate the sequence of partial sums from an original sequence. Find a specific term of a sequence (a_n) given its partial sums (S_n and S_{n-1}). Apply formulas for the sum of finite arithmetic and geometric sequences to find partial sums efficiently. If you stack 1 can on the top row of a display, 2 on the second, 3 on the third, and so on, how many total cans are in a 10-row display? 🥫🤔 This lesson introduces the concept of partial sums, which are the sums of the first 'n' terms of a sequence. Understanding partial sums is the crucia...

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific pattern or rule. We denote the nth term as a_n.The sequence of even positive integers is {2, 4, 6, 8, ...}, where the general term is a_n = 2n. Term of a Sequence (a_n)A single element or number at a specific position 'n' within a sequence.In the sequence {2, 4, 6, 8, ...}, the 3rd term is a_3 = 6. SeriesThe sum of all the terms in a sequence. A series can be finite or infinite.For the sequence {2, 4, 6, 8, ...}, the corresponding series is 2 + 4 + 6 + 8 + ... Partial Sum (S_n)The sum of a specified number of terms from the beginning of a sequence. S_n is the sum of the first 'n' terms.For the sequence {2, 4, 6, 8, ...}, the 3rd partial sum is S_3 = a_1 + a_2 + a_3 = 2 + 4 +...

The nth Partial Sum (Definition) S_n = a_1 + a_2 + a_3 + ... + a_n This is the fundamental definition. To find the nth partial sum, you add up all the terms of the sequence from the first term (a_1) up to the nth term (a_n). The nth Partial Sum (Summation Notation) S_n = \sum_{k=1}^{n} a_k This is a compact way to write the partial sum. The Greek letter Sigma (Σ) means 'sum'. This notation reads 'the sum of a_k as k goes from 1 to n'. Finding a Term from Partial Sums a_n = S_n - S_{n-1} \text{ (for n > 1)} The nth term of a sequence is the difference between the nth partial sum and the previous partial sum. Note that a_1 = S_1.