Describe the error in finding the sum. ∑k=1^4(3+2 k^2) =∑k=1^4 3+∑k=1^4 2 k^2 =3+(2+8+1

Name: Problem 105, Chapter 9: Precalculus
Uploaded: 2021-02-07T14:40:26-08:00
Duration: 2 min 53 s
Channel: Anurag Kumar
Description: Describe the error in finding the sum. ∑k=1^4(3+2 k^2) =∑k=1^4 3+∑k=1^4 2 k^2 =3+(2+8+18+32) =63

step 1 In this problem of series, we have given that this is given that summation of this is 3 plus 3K

Question

Describe the error in finding the sum. $$\begin{aligned} \sum_{k=1}^{4}\left(3+2 k^{2}\right) &=\sum_{k=1}^{4} 3+\sum_{k=1}^{4} 2 k^{2} \\ &=3+(2+8+18+32) \\ &=63 \end{aligned}$$

   Describe the error in finding the sum.
$$\begin{aligned}
\sum_{k=1}^{4}\left(3+2 k^{2}\right) &=\sum_{k=1}^{4} 3+\sum_{k=1}^{4} 2 k^{2} \\
&=3+(2+8+18+32) \\
&=63
\end{aligned}$$

Precalculus

Ron Larson 10th Edition

Chapter 9, Problem 105

Instant Answer

Solved by Expert Anurag Kumar

02/07/2021

Step 1

The sum of a constant term over a range is simply the constant multiplied by the number of terms in the range. In this case, the range is from 1 to 4, so there are 4 terms. Therefore, the sum of the constant term 3 is $3 \times 4 = 12$. Show more…

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Describe the error in finding the sum. $$\begin{aligned} \sum_{k=1}^{4}\left(3+2 k^{2}\right) &=\sum_{k=1}^{4} 3+\sum_{k=1}^{4} 2 k^{2} \\ &=3+(2+8+18+32) \\ &=63 \end{aligned}$$

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Key Concepts

Summation Notation

Summation notation is a shorthand used to represent the addition of a sequence of terms. It involves an index of summation that runs from a lower bound to an upper bound, allowing for concise and clear representation of series and sums.

Linearity of Summation

The linearity of summation is the principle that the sum of a linear combination of functions is equal to the same linear combination of their sums. Mathematically, this means that ?(f(k) + g(k)) = ?f(k) + ?g(k), provided that the summation limits are the same. This property must be applied carefully to ensure that each term is correctly handled.

Summing a Constant

When summing a constant over an index, the constant must be multiplied by the number of terms in the summation. For example, ? from k = 1 to n of a constant c is equal to c multiplied by n. A common error is to treat the constant as appearing only once in the sum, rather than accounting for all occurrences across the index range.

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