91Ó°ÊÓ

Evaluating a series means finding the sum of a sequence of numbers, often expressed with summation notation. In the given exercise, we have the sum oindent \ oindent $$ \sum_{k=1}^{7}(-1)^{k} 4^{k+1}$$. oindent \ oindent This notation tells us to evaluate the expression \ oindent $$\begin{cases} (-1)^{k} 4^{k+1} \end{cases} $$ for each integer value of \(k\) from 1 to 7 and then sum these results. oindent \ oindent To achieve this, follow these steps: oindent \ oindent

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• Substitute each value of \(k\) into the expression
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• Evaluate the expression for each value of \(k\)
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• Add all the resulting terms together

oindent \ oindent By systematically substituting the values and performing each calculation, you’ll get the final sum for the series. This structured approach helps students understand the process and reduces the chance for errors.

Exponentiation is an operation where a number, called the base, is multiplied by itself a certain number of times indicated by the exponent. In simpler terms, \(a^b\) means that \(a\) (the base) is multiplied by itself \(b\) times. oindent \ oindent For instance, in the given series oindent oindent $$\sum_{k=1}^{7}(-1)^{k} 4^{k+1},$$ oindent \ oindent we see \(4^{k+1}\) where 4 is the base, and \(k+1\) is the exponent. As we substitute each \(k\) value, we must perform exponentiation: oindent \ oindent

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• When \(k=1\), \(4^{1+1}=4^2=16\)
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• When \(k=2\), \(4^{2+1}=4^3=64\)
• When \(k=3\), \(4^{3+1}=4^4=256\)

oindent \ oindent Proper understanding of exponentiation is crucial for accurately evaluating the terms of the series. Remember, exponentiation grows very quickly; small changes in the exponent lead to large changes in the result. In this problem, the exponential growth is evident as values of \(k\) increase.

In mathematics, an alternating sequence is one where the signs of the terms switch back and forth between positive and negative. In the provided series oindent \ oindent $$\sum_{k=1}^{7}(-1)^{k} 4^{k+1},$$ oindent \ oindent the term \ oindent \((-1)^{k}\) oindent forces the sequence to alternate: oindent oindent \ oindent

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• When \(k\) is odd, \((-1)^{k}\) is negative
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• When \(k\) is even, \((-1)^{k}\) is positive

oindent \ oindent This mechanism ensures that the series flips signs at each step, affecting the overall sum by alternating between adding and subtracting subsequent terms. oindent \ oindent For example: oindent

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• When \(k=1\), \((-1)^1 = -1\), so \(-4^2 = -16\)
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• When \(k=2\), \((-1)^2 = 1\), so \(1 * 4^3 = 64\)

oindent \ oindent Observing and understanding this alternating nature helps us predict the behavior of the series and accurately compute the final sum.