- #1
Albert1
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$$\sum_{k=1}^{99}\dfrac {(2k+1)+\sqrt{k(k+1)}}{\sqrt k+\sqrt{k+1}}$$
Albert said:$$\sum_{k=1}^{99}\dfrac {(2k+1)+\sqrt{k(k+1)}}{\sqrt k+\sqrt{k+1}}$$
The symbol ∑ represents the sum of a series of terms. In this equation, it indicates that we are adding up all the values of the expression within the parentheses for each value of k from 1 to n.
To simplify this equation, we can start by breaking down the numerator and denominator separately. For the numerator, we can use the distributive property to expand the expression (2k+1)+√k(k+[1]). Then, we can combine like terms and simplify further. For the denominator, we can use the property of square roots to simplify the expression (√k+√(k+1)). Finally, we can divide the simplified numerator by the simplified denominator to get our final answer.
The square root is important in this equation because it is used to simplify the denominator. By combining two square roots, we can eliminate the radical symbol and simplify the expression. The square root also plays a role in expanding the expression in the numerator, as shown in the previous question.
In this equation, k can take on any positive integer value from 1 to n. This is because the ∑ symbol indicates that we are adding up the expression for each value of k within that range. If the range is not specified, then k can take on any positive integer value.
The purpose of using summation notation in this equation is to represent a series of terms that follow a specific pattern. Instead of writing out each individual term, we can use the ∑ symbol to indicate that we are adding up all the terms from k=1 to n. This notation makes it easier to work with and understand formulas that involve a large number of terms.