Evaluate the Summation sum from x=2 to 7 of (x+1)/x
This question is asking for the evaluation of a finite summation. Specifically, the task is to calculate the sum of the terms (x+1)/x as x ranges from the integer 2 up to 7. For each value of x within this range, you would calculate the expression (x+1)/x, which is a rational expression, and then add all these calculated values together to find the total sum.
$\sum_{x = 2}^{7} \frac{x + 1}{x}$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer $x$ from 2 to 7.
$1 + \frac{1}{2} + 1 + \frac{1}{3} + 1 + \frac{1}{4} + 1 + \frac{1}{5} + 1 + \frac{1}{6} + 1 + \frac{1}{7}$
Step 2:
Combine the terms to find the sum.
$\frac{1063}{140}$
Step 3:
Express the sum in various numerical formats.
- Exact Form: $\frac{1063}{140}$
- Decimal Form: $7.59285714\ldots$
- Mixed Number Form: $7 \frac{83}{140}$
Solution:"The summation of the series from x=2 to 7 of (x+1)/x is calculated by expanding the series, simplifying the terms, and then expressing the result in various forms. The exact sum is $\frac{1063}{140}$, which can also be written as a decimal $7.59285714\ldots$ or as a mixed number $7 \frac{83}{140}$."
Knowledge Notes:
Summation notation $\sum$ is used to denote the addition of a sequence of numbers.
The series $(x+1)/x$ can be expanded by substituting each integer value of $x$ within the given range into the formula.
Simplifying the series involves combining like terms and performing any necessary arithmetic operations.
The result of a summation can be expressed in different numerical forms, such as an exact fraction, a decimal approximation, or a mixed number.
A mixed number consists of an integer part and a fractional part, representing the sum as a whole number plus a fraction.
