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Magazine Chapter 13: Problem 56
Rewrite the sums using sigma notation. $$x+2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+6 x^{6}$$
Short Answer
Expert verified
The expression can be rewritten as \( \sum_{n=1}^{6} n \cdot x^n \).
Step by step solution
01
Understanding the Problem
We are given the expression \(x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6\). We need to recognize a pattern within the terms that will allow us to write this sum using Sigma (\(\Sigma\)) notation.
02
Identifying the Pattern
Observing the terms, we see that the expression is a polynomial where each term consists of a coefficient that increases linearly (i.e., 1, 2, 3, 4, 5, 6) multiplying a power of \(x\) that starts at \(1\) and increases as well (i.e., \(x^1, x^2, x^3, \ldots, x^6\)).
03
Defining the General Term
In general, for the nth term, the pattern can be expressed as \(n \cdot x^n\). The terms start from \(x\) for \(n = 1\) and go up to \(6x^6\) for \(n = 6\).
04
Writing in Sigma Notation
Using the pattern identified, we can rewrite the given polynomial as a sum of terms. The sigma notation for this expression is: \[\sum_{n=1}^{6} n \cdot x^n .\]
05
Verification
We can verify by expanding \(\sum_{n=1}^{6} n \cdot x^n\) to ensure it aligns with each term of the original expression:- For \(n=1\), the term is \(1 \cdot x = x\).- For \(n=2\), the term is \(2 \cdot x^2\).- For \(n=3\), the term is \(3 \cdot x^3\), and so on up to \(n=6\). The sigma notation expression matches the given polynomial.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Expression
A polynomial expression is a mathematical expression that consists of variables, coefficients, and exponents. In our example, the polynomial is made up of terms like \(x, 2x^2, 3x^3\), and so on. Each term in a polynomial consists of:
- A variable part, which in this case is \(x\).
- A coefficient, like 1, 2, 3, etc., which is the numerical factor.
- An exponent that indicates the power to which the variable is raised, such as \(x^1, x^2, x^3\).
Pattern Recognition
Pattern recognition plays a crucial role in understanding and simplifying mathematical expressions. In the context of this problem, pattern recognition involves identifying a consistent structure among the terms of the polynomial.
For the expression provided, observe that:
For the expression provided, observe that:
- The coefficients are consecutive natural numbers: 1, 2, 3, 4, 5, and 6.
- The powers of \(x\) also increase steadily from 1 to 6.
General Term
The general term in a pattern or sequence is a formula that represents any term in the sequence. Identifying this helps in working with sigma notation.
In our example, the general term is \( n \cdot x^n \). This formulation comes from how each term combines its position in the sequence \(n\), as a coefficient with the power \(x^n\):
In our example, the general term is \( n \cdot x^n \). This formulation comes from how each term combines its position in the sequence \(n\), as a coefficient with the power \(x^n\):
- The first term \(n = 1\) results in \(1 \cdot x^1\).
- The second term \(n = 2\) results in \(2 \cdot x^2\).
- This pattern continues through to the sixth term \(6 \cdot x^6\).
Verification of Expression
Verification is an essential step to ensure that the rewritten expression accurately reflects the original expression. In the case of this polynomial, it involves checking each term in the sigma notation expansion matches the initial polynomial terms.
When we expand \( \sum_{n=1}^{6} n \cdot x^n \), each term corresponds perfectly to those in our given sequence:
When we expand \( \sum_{n=1}^{6} n \cdot x^n \), each term corresponds perfectly to those in our given sequence:
- \(n = 1\) gives \(x\).
- \(n = 2\) gives \(2x^2\).
- This pattern continues up to \(n = 6\), providing \(6x^6\).
