Partial Sums & Summation 12/9/24

Front

What is a partial sum?

Back

A partial sum is the sum of the first 'n' terms of a sequence. It is denoted as S_n, where n indicates the number of terms being summed.

Front

How do you denote the sum of a sequence using sigma notation?

Back

Sigma notation uses the Greek letter sigma (Σ) to represent the sum of a sequence. For example, the sum of the first n terms of a sequence a_k can be written as Σ_{k=1}^{n} a_k.

Front

Evaluate the partial sum S_3 for the sequence a_n = 3n^2 - 8.

Back

S_3 = a_1 + a_2 + a_3 = (3(1)^2 - 8) + (3(2)^2 - 8) + (3(3)^2 - 8) = -5 + 4 + 13 = 18.

Front

What is the formula for the sum of the first n squares?

Back

The formula for the sum of the first n squares is S_n = \frac{n(n + 1)(2n + 1)}{6}.

Front

Write the series 1 + 4 + 9 + ... + 49 using sigma notation.

Back

Front

What is the sum of the series 1 + 2 + 3 + ... + n?

Back

The sum of the series is given by the formula S_n = \frac{n(n + 1)}{2}.

Front

How do you find the sum of a finite arithmetic series?

Back

The sum of a finite arithmetic series can be found using the formula S_n = \frac{n}{2} (a_1 + a_n), where a_1 is the first term, a_n is the last term, and n is the number of terms.

Front

What is the difference between a sequence and a series?

Back

A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.

Front

Evaluate the sum of the series 1 + 3 + 5 + ... + 19.

Back

The sum can be calculated as S_n = n^2, where n is the number of terms. Here, n = 10, so S_n = 10^2 = 100.

Front

What is the nth term of the sequence defined by a_n = 3n^2 - 8?

Back

The nth term is given by substituting n into the formula: a_n = 3n^2 - 8.