- #1

KingNothing

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Is there a nice formula for calculating the partial sum of the series n^2 from 1 to k?

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In summary, the formula for calculating the partial sum of n^2 from 1 to k is S_k = k(k+1)(2k+1)/6. This formula is derived using the sum of squares formula and can be used for any positive integer value of k. The partial sum of n^2 from 1 to k represents the sum of the squares of the first k natural numbers and has applications in various fields of mathematics and statistics. This formula can be applied in real-world situations, such as calculating costs, temperatures, and population estimates.

- #1

KingNothing

- 881

- 4

Is there a nice formula for calculating the partial sum of the series n^2 from 1 to k?

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- #2

quasar987

Science Advisor

Homework Helper

Gold Member

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k(k+1)(2k+1)/6

- #3

StatusX

Homework Helper

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You can derive that as follows:

[tex]k^3=\sum_{n=1}^k (n^3-(n-1)^3)=\sum_{n=1}^k (n^3-(n^3-3n^3+3n-1))=3\sum_{n=1}^k n^2-3\sum_{n=1}^k n+\sum_{n=1}^k 1=3\sum_{n=1}^k n^2-3k(k+1)/2+k[/tex]

[tex]\sum_{n=1}^k n^2=\frac{1}{3} (k^3+3k(k+1)/2-k)=\frac{1}{3}(k(k+1)(k-1)+3k(k+1)/2))=k(k+1)(k+1/2)/3=k(k+1)(2k+1)/6[/tex]

[tex]k^3=\sum_{n=1}^k (n^3-(n-1)^3)=\sum_{n=1}^k (n^3-(n^3-3n^3+3n-1))=3\sum_{n=1}^k n^2-3\sum_{n=1}^k n+\sum_{n=1}^k 1=3\sum_{n=1}^k n^2-3k(k+1)/2+k[/tex]

[tex]\sum_{n=1}^k n^2=\frac{1}{3} (k^3+3k(k+1)/2-k)=\frac{1}{3}(k(k+1)(k-1)+3k(k+1)/2))=k(k+1)(k+1/2)/3=k(k+1)(2k+1)/6[/tex]

Last edited:

The formula for calculating the partial sum of n^2 from 1 to k is given by:

S_{k} = k(k+1)(2k+1)/6

This formula is derived using the sum of squares formula, which states that the sum of squares from 1 to n is equal to n(n+1)(2n+1)/6. By replacing n with k, we get the formula for the partial sum of n^2 from 1 to k.

Yes, this formula can be used for any positive integer value of k.

The partial sum of n^2 from 1 to k represents the sum of the squares of the first k natural numbers. It has applications in various fields of mathematics and statistics, such as in calculating areas under curves and in statistical analysis.

This formula can be applied in various real-world situations, such as calculating the total cost of goods sold over a period of time, determining the average temperature over a certain time period, or estimating the total number of cells in a growing population.

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