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Chapter 13: Problem 17

List all terms of each finite sequence. \(c_{n}=n^{-1 / 2}\) for \(1 \leq n \leq 5\)

Short Answer

Expert verified
1, 0.707, 0.577, 0.5, 0.447

Step by step solution

01

Understand the Sequence Formula

The sequence is given by the formula: \[ c_{n} = n^{-1/2} \] This means that for each term in the sequence, take the value of n, raise it to the power of -1/2.
02

Calculate the First Term

Substitute \( n = 1 \) into the formula: \[ c_{1} = 1^{-1/2} = 1 \]
03

Calculate the Second Term

Substitute \( n = 2 \) into the formula: \[ c_{2} = 2^{-1/2} = \frac{1}{\sqrt{2}} \approx 0.707 \]
04

Calculate the Third Term

Substitute \( n = 3 \) into the formula: \[ c_{3} = 3^{-1/2} = \frac{1}{\sqrt{3}} \approx 0.577 \]
05

Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \[ c_{4} = 4^{-1/2} = \frac{1}{2} = 0.5 \]
06

Calculate the Fifth Term

Substitute \( n = 5 \) into the formula: \[ c_{5} = 5^{-1/2} = \frac{1}{\sqrt{5}} \approx 0.447 \]

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Key Concepts

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

sequence formula
A sequence formula is a mathematical expression that defines the nth term of a sequence using the index number n. In our problem, the formula is given by \( c_{n} = n^{-1/2} \). This means each term in the sequence is determined by plugging different values of n into the formula. The sequence starts from n = 1 and goes up to n = 5. Always remember to understand the formula first. This will make term calculation so much easier.
term calculation
Term calculation involves substituting the integer values of the sequence index (n) into the sequence formula to find each specific term. Here's how to calculate the terms in our example sequence:
inverse powers
Inverse powers refer to numbers raised to a negative exponent, like \(n^{-1/2}\). This is the reciprocal of the positive power. For example, \(2^{-1/2} = \frac{1}{\sqrt{2}}\). Understanding inverse powers is very important for calculating the terms in our sequence.

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