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Chapter 14: Problem 53

Find the indicated partial sum for each sequence. $$ 2,4,6,8, \dots ; S_{5} $$

Short Answer

Expert verified
The partial sum \(S_5\) of the sequence is 30.

Step by step solution

01

Identify the sequence type

The given sequence is an arithmetic sequence with the first term, \(a = 2\), and the common difference, \(d = 2\).
02

Use the formula for the nth term of an arithmetic sequence

The nth term of an arithmetic sequence can be found using the formula: \(a_n = a + (n-1)d\). For the 5th term, \(a_5 = 2 + (5-1)2 = 2 + 8 = 10\).
03

Use the formula for the sum of the first n terms

The sum of the first n terms of an arithmetic sequence is given by: \(S_n = \frac{n}{2} (a + a_n)\). For \(S_5\): \(S_5 = \frac{5}{2}(2 + 10) = \frac{5}{2} \times 12 = 5 \times 6 = 30\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Sequence
In mathematics, an arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This is known as the common difference, denoted by \(d\). For instance, in the sequence given in the problem: \(2, 4, 6, 8, \text{...}\), the common difference is \(d = 2\). Every subsequent term is obtained by adding the common difference to the previous term.
Nth Term Formula
The nth term of an arithmetic sequence can be determined using the formula: \(a_n = a + (n-1)d\). Here, \(a_n\) represents the nth term, \(a\) is the first term, and \(d\) is the common difference. For example, to find the 5th term of the sequence \(2,4,6,8,...\), substitute the values into the formula. This gives us: \(a_5 = 2 + (5-1)2 = 2 + 8 = 10\).
Sum of First n Terms
To find the sum of the first \(n\) terms of an arithmetic sequence, we use the formula: \(S_n = \frac{n}{2} (a + a_n)\). \(S_n\) stands for the sum of the first \(n\) terms, \(a\) is the first term, and \(a_n\) is the nth term. For the sequence in the example, we need to find \(S_5\): Plugging in the values, we get: \(S_5 = \frac{5}{2}(2 + 10) = \frac{5}{2} \times 12 = 5 \times 6 = 30\). Thus, the sum of the first 5 terms is 30.

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Most popular questions from this chapter

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