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\( \displaystyle\sum_{k = 1}^{n}{a_k} = a_1 + a_2 + a_3 + \ldots + a_n \)

ÖRNEK:

\( \displaystyle\sum_{k = 1}^{10}{a^2_k} = a^2_1 + a^2_2 + \ldots + a^2_{10} \)

\( \displaystyle\sum_{k = -2}^{2}{f(k)} = f(-2) + f(-1) \) \( + f(0) + f(1) + f(2) \)

\( \displaystyle\sum_{k = 1}^{n}(a_k \pm b_k) \) \( = \displaystyle\sum_{k = 1}^{n}{a_k} \pm \displaystyle\sum_{k = 1}^{n}{b_k} \)

\( \displaystyle\sum_{k = 1}^{n}(a_k \cdot b_k) \) \( \ne \displaystyle\sum_{k = 1}^{n}{a_k} \cdot \displaystyle\sum_{k = 1}^{n}{b_k} \)

\( \displaystyle\sum_{k = 1}^{n}(c \cdot a_k) \) \( = c \cdot \displaystyle\sum_{k = 1}^{n}{a_k} \)

\( 1 \lt j \lt n \) olmak üzere,

\( \displaystyle\sum_{k = 1}^{n}{a_k} \) \( = \displaystyle\sum_{k = 1}^{j}{a_k} + \displaystyle\sum_{k = j + 1}^{n}{a_k} \)

ÖRNEK:

\( \displaystyle\sum_{k = 1}^{50}{k^2} \) \( = \displaystyle\sum_{k = 1}^{20}{k^2} + \displaystyle\sum_{k = 21}^{50}{k^2} \)

\( \displaystyle\sum_{k = 1}^{n}{k} \) \( = 1 + 2 + 3 + \ldots + n \) \( = \dfrac{n \cdot (n + 1)}{2} \)

\( \displaystyle\sum_{k = 1}^{n}{2k} \) \( = 2 + 4 + 6 + \ldots + 2n \) \( = n \cdot (n + 1) \)

\( \displaystyle\sum_{k = 1}^{n}(2k - 1) \) \( = 1 + 3 + 5 + \ldots + (2n - 1) \) \( = n^2 \)

\( \displaystyle\sum_{k = 1}^{n}(2k - 1)^2 \) \( = 1^2 + 3^2 + 5^2 + \ldots + (2n - 1)^2 \) \( = \dfrac{n \cdot (4n^2 - 1)}{3} \)

\( r \ne 1 \) olmak üzere,

\( \displaystyle\sum_{k = 1}^{n}{r^{k - 1}} \) \( = 1 + r + r^2 + \ldots + r^{n - 1} \) \( = \dfrac{1 - r^n}{1 - r} \)

\( \displaystyle\sum_{k = 1}^{n}{k^2} \) \( = 1^2 + 2^2 + 3^2 + \ldots + n^2 \) \( = \dfrac{n \cdot (n + 1) \cdot (2n + 1)}{6} \)

\( \displaystyle\sum_{k = 1}^{n}{k^3} \) \( = 1^3 + 2^3 + 3^3 + \ldots + n^3 \) \( = {\left[ \dfrac{n \cdot (n + 1)}{2} \right]}^2 \)

\( \displaystyle\sum_{k = 1}^{n}{k \cdot (k + 1)} \) \( = 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots \) \( + n \cdot (n + 1) \) \( = \dfrac{n \cdot (n + 1) \cdot (n + 2)}{3} \)

\( \displaystyle\sum_{k = 1}^{n}{\dfrac{1}{k \cdot (k + 1)}} \) \( = \dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + \ldots \) \( + \dfrac{1}{n \cdot (n + 1)} \) \( = \dfrac{n}{n + 1} \)

\( \displaystyle\sum_{k = 1}^{n}{k \cdot k!} \) \( = 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \ldots \) \( + n \cdot n! \) \( = (n + 1)! - 1 \)

SORU 1:

\( (a_n) = \dfrac{1}{\sqrt{n+2} + \sqrt{n+1}} \) olduğuna göre,

\( \displaystyle\sum_{k = 1}^8{a_k} \) toplamının sonucu kaçtır?