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Chapter 13: Problem 14

Write the first five terms of each sequence. $$ a_{n}=-\frac{2}{n^{2}} $$

Short Answer

Expert verified
The first five terms are -2, -0.5, -0.22, -0.125, and -0.08.

Step by step solution

01

Identify the Sequence Formula

The given formula for the sequence is \( a_{n} = -\frac{2}{n^{2}} \).
02

Calculate the First Term

Substitute \( n = 1 \) into the sequence formula: \( a_{1} = -\frac{2}{1^{2}} = -2 \).
03

Calculate the Second Term

Substitute \( n = 2 \) into the sequence formula: \( a_{2} = -\frac{2}{2^{2}} = -\frac{2}{4} = -\frac{1}{2} \).
04

Calculate the Third Term

Substitute \( n = 3 \) into the sequence formula: \( a_{3} = -\frac{2}{3^{2}} = -\frac{2}{9} \).
05

Calculate the Fourth Term

Substitute \( n = 4 \) into the sequence formula: \( a_{4} = -\frac{2}{4^{2}} = -\frac{1}{8} \).
06

Calculate the Fifth Term

Substitute \( n = 5 \) into the sequence formula: \( a_{5} = -\frac{2}{5^{2}} = -\frac{2}{25} \).

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Key Concepts

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

Term Calculation
When we talk about term calculation in a sequence, we determine each individual term using a given formula. In the sequence provided, the formula is: \( a_n = -\frac{2}{n^2} \). This formula allows us to find any term in the sequence once we know the position of that term, denoted as \( n \). To find the term at a specific position:
  • Substitute the position number (\( n \)) into the formula
  • Perform the arithmetic operations to simplify the expression
Let's break it down further with our sequence:
  • For the 1st term: \( a_{1} = -\frac{2}{1^2} = -2 \)
  • For the 2nd term: \( a_{2} = -\frac{2}{2^2} = -\frac{1}{2} \)
  • For the 3rd term: \( a_{3} = -\frac{2}{3^2} = -\frac{2}{9} \)
  • For the 4th term: \( a_{4} = -\frac{2}{4^2} = -\frac{1}{8} \)
  • For the 5th term: \( a_{5} = -\frac{2}{5^2} = -\frac{2}{25} \)
Numerical Sequences
Numerical sequences are ordered lists of numbers defined by a particular rule or formula. The terms in a numerical sequence follow this given rule to ensure consistency. For example, in our sequence formula, \( a_n = -\frac{2}{n^2} \), every term is generated by substituting the position number into the sequence formula. Numerical sequences can be:
  • Arithmetic, where the difference between consecutive terms is constant
  • Geometric, where the ratio of consecutive terms is constant
  • Based on other rules or formulas, like the one in our example
The sequence given is neither arithmetic nor geometric鈥攊t is determined by the formula \( a_n = -\frac{2}{n^2} \), which governs how each term is calculated.
Substitution Method
The substitution method involves replacing a variable with a specific value to calculate an expression. This method is crucial in finding terms of a sequence from a formula. Here's how to use the substitution method to find sequence terms:
  • Identify the formula for the sequence
  • Substitute the desired term's position number (\( n \)) into the formula
  • Simplify the resulting expression to find the term
Let's apply it step-by-step on our sequence formula \( a_n = -\frac{2}{n^2} \):
  • To find the first term, we substitute \( n = 1 \): \( a_{1} = -\frac{2}{1^2} = -2 \)
  • To find the second term, substitute \( n = 2 \): \( a_{2} = -\frac{2}{2^2} = -\frac{1}{2} \)
  • Continuing this for \( n = 3 \), \( n = 4 \), and \( n = 5 \) gives us the corresponding terms
By using substitution, we can systematically find any term in the sequence by plugging in the appropriate value for \( n \).

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