Express the following sum in closed form. ∑ k=1. 3(k/n) =?

Express the following sum in closed form. Your answer should be in terms of n.

1. ∑ k=1. 3(k/n) = ?

2. ∑ k=1. (4+2k)^2 = ?

3. ∑ k=1. (3+3* k/n)^2= ?

1 Answer

  • 1.

    ∑ k=1. 3(k/n)

    = (3/n)∑ k=1. (k)

    = (3/n)(n/2)(n - 1)

    = (3/2)(n - 1)

    2.

    ∑ k=1. (4+2k)^2

    = ∑ k=1. (16 + 16k + 4k^2)

    = 16n + 16(n/2)(n - 1) + 4[(1/3)n^3 + (1/2)n^2 + (1/6)n]

    = 16n + (8n^2 - 8n) + [(4/3)n^3 + 2n^2 + (2/3)n]

    = (4/3)n^3 + 10n^2 + (26/3)n

    3.

    ∑ k=1. (3+3*k/n)^2

    = ∑ k=1. (9 + 18*k/n + 9k^2/n^2)

    = 9n + (18/n)(n/2)(n - 1) + (9/n^2)[(1/3)n^3 + (1/2)n^2 + (1/6)n]

    = 9n + (9n - 9) + [3n + (9/2) + (3/2n)]

    = 21n - 9/2 + 3/(2n)

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