Evaluate the Summation sum from m=1 to 25 of 50-m
This problem asks for the evaluation of a mathematical summation. Specifically, you need to calculate the sum of a series of numbers generated by the expression (50 - m), where m takes on all integer values from 1 to 25, inclusive. You would do this by iterating m through those values and summing the results of the expression for each m.
$\sum_{m = 1}^{25} 50 - m$
Answer
Solution:
Step 1:
Decompose the given summation into two separate summations that are easier to handle. We have:
$$\sum_{m = 1}^{25} (50 - m) = \sum_{m = 1}^{25} 50 - \sum_{m = 1}^{25} m$$
Step 2:
Calculate the summation of the constant term $50$ over the range from $1$ to $25$.
Step 2.1:
The summation of a constant $c$ over $n$ terms is given by:
$$\sum_{k = 1}^{n} c = c \cdot n$$
Step 2.2:
Insert the known values into the formula:
$$(50) \cdot (25)$$
Step 2.3:
Perform the multiplication of $50$ times $25$ to get:
$$1250$$
Step 3:
Now, compute the summation of the sequence of first $25$ natural numbers.
Step 3.1:
The formula for the summation of the first $n$ natural numbers is:
$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$
Step 3.2:
Plug in the values into the formula, remembering to include the negative sign from the original summation:
$$(-1) \cdot \left(\frac{25(25 + 1)}{2}\right)$$
Step 3.3:
Simplify the expression.
Step 3.3.1:
Add $25$ and $1$ together:
$$-1 \cdot \frac{25 \cdot 26}{2}$$
Step 3.3.2:
Multiply $25$ by $26$ to find the product:
$$-1 \cdot \left(\frac{650}{2}\right)$$
Step 3.3.3:
Divide $650$ by $2$ to get:
$$-1 \cdot 325$$
Step 3.3.4:
Finally, multiply $-1$ by $325$ to obtain:
$$-325$$
Step 4:
Combine the results from the two summations calculated in steps 2 and 3.
Step 5:
Subtract $325$ from $1250$ to get the final result:
$$925$$
Knowledge Notes:
Summation of a Constant: The sum of a constant $c$ repeated $n$ times is simply $c$ multiplied by $n$.
Summation of Natural Numbers: The sum of the first $n$ natural numbers is given by the formula $\frac{n(n + 1)}{2}$. This is a result of the arithmetic series where the difference between consecutive terms is constant.
Linearity of Summation: The summation operator is linear, which means that the sum of a sequence of terms can be split into the sum of two sequences of terms, as long as the sequences do not depend on each other.
Negative Signs in Summation: When dealing with negative signs in summation, it is important to distribute the negative sign across the terms being summed.
Combining Summations: After evaluating individual summations, their results can be combined (added or subtracted) according to the original problem to find the final sum.
