Sigma Q: sum[k=1,4]5k^2 (don't understand their solution ?)

[ManolitoAguirre]

Q. What is the value of \(\displaystyle \displaystyle \sum_{k=1}^4\, 5k^2\,\)?

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I guessed the answer, but I was wondering about their solution:

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A. Use the formula

. . .\(\displaystyle \displaystyle \sum_{k=1}^n\, k^2\, =\, \dfrac{1}{6}\, n\, (n\, +\, 1)\, (2n\, +\, 1)\)

Therefore

. . .\(\displaystyle \displaystyle \sum_{k=1}^4\, 5k^2\, =\, 5\, \sum_{k=1}^4\, k^2\)

. . . . .\(\displaystyle \displaystyle =\, 5\,\times\, \dfrac{1}{6}\, \times\, 4\, \times\, 5\, \times\, 9\, =\, 150\)

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In the question, where did 1/6 come from?

Thank you

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[Deleted member 4993]

Q. What is the value of \(\displaystyle \displaystyle \sum_{k=1}^4\, 5k^2\,\)?

---

I guessed the answer, but I was wondering about their solution:

---

A. Use the formula

. . .\(\displaystyle \displaystyle \sum_{k=1}^n\, k^2\, =\, \dfrac{1}{6}\, n\, (n\, +\, 1)\, (2n\, +\, 1)\)

Therefore

. . .\(\displaystyle \displaystyle \sum_{k=1}^4\, 5k^2\, =\, 5\, \sum_{k=1}^4\, k^2\)

. . . . .\(\displaystyle \displaystyle =\, 5\,\times\, \dfrac{1}{6}\, \times\, 4\, \times\, 5\, \times\, 9\, =\, 150\)

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In the question, where did 1/6 come from?

Thank you

From knowing a-priori:

1 + 2² + 3² + 4².....+ (n-1)² + n² = 1/6 * n * (n+1) * (2n+1)

Do a google-search on "sum of square series" for detailed method.

 

[stapel]

Q. What is the value of \(\displaystyle \displaystyle \sum_{k=1}^4\, 5k^2\,\)?

---

I guessed the answer, but I was wondering about their solution:

---

A. Use the formula

. . .\(\displaystyle \displaystyle \sum_{k=1}^n\, k^2\, =\, \dfrac{1}{6}\, n\, (n\, +\, 1)\, (2n\, +\, 1)\)

Therefore

. . .\(\displaystyle \displaystyle \sum_{k=1}^4\, 5k^2\, =\, 5\, \sum_{k=1}^4\, k^2\)

. . . . .\(\displaystyle \displaystyle =\, 5\,\times\, \dfrac{1}{6}\, \times\, 4\, \times\, 5\, \times\, 9\, =\, 150\)

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In the question, where did 1/6 come from?

It's just part of the formula they gave you. Either they proved the formula (maybe in the section on induction proofs), or they just gave it to you to use (and perhaps memorize for the next test).