Evaluate the Summation sum from i=1 to 72 of i/4
The question asks you to compute the value of a specific mathematical summation. The summation is over the variable $i$, which ranges from 1 to 72. For each value of $i$within this range, you are to divide $i$by 4, and then add up all these individual fractions. Essentially, the question wants you to calculate the accumulated sum of the first 72 positive integers, each divided by 4.
$\sum_{i = 1}^{72} \frac{i}{4}$
Answer
Solution:
Step 1:
Extract the constant $\frac{1}{4}$ from the summation: $\frac{1}{4} \sum_{i = 1}^{72} i$
Step 2:
Utilize the summation formula for the first $n$ natural numbers: $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$
Step 3:
Insert the upper limit of the summation into the formula, and don't forget to include the constant we factored out: $\frac{1}{4} \left( \frac{72(72 + 1)}{2} \right)$
Step 4:
Proceed with the simplification process.
Step 4.1:
Begin by simplifying the numerator.
Step 4.1.1:
Combine $72$ and $1$: $\frac{1}{4} \cdot \frac{72 \cdot 73}{2}$
Step 4.1.2:
Perform the multiplication of $72$ by $73$: $\frac{1}{4} \cdot \frac{5256}{2}$
Step 4.2:
Eliminate the common factor of $4$.
Step 4.2.1:
Extract $4$ from the numerator: $\frac{1}{4} \cdot \frac{4 \cdot 1314}{2}$
Step 4.2.2:
Reduce the common factors: $\frac{1}{\cancel{4}} \cdot \frac{\cancel{4} \cdot 1314}{2}$
Step 4.2.3:
Rewrite the simplified expression: $\frac{1314}{2}$
Step 4.3:
Finish by dividing $1314$ by $2$: $657$
Knowledge Notes:
The problem involves evaluating a summation of a sequence of numbers, specifically the first 72 natural numbers divided by 4. The process of solving this problem involves several key knowledge points:
Summation Notation: The summation notation $\sum$ is used to denote the addition of a sequence of numbers. The expression $\sum_{i=1}^{n} a_i$ means to sum the sequence $a_i$ from $i=1$ to $i=n$.
Factoring Constants from Summations: When a constant is multiplied by each term in a summation, it can be factored out to simplify the expression. This is based on the distributive property of multiplication over addition.
Summation Formula for Natural Numbers: The formula for the sum of the first $n$ natural numbers is given by $\sum_{i=1}^{n} i = \frac{n(n + 1)}{2}$. This formula is derived from the observation that pairing the first and last terms, second and second-to-last terms, etc., each pair sums to $n+1$, and there are $\frac{n}{2}$ such pairs.
Simplification: The process of simplification involves performing arithmetic operations to reduce an expression to its simplest form. This includes adding, multiplying, and canceling out common factors.
Arithmetic Operations: Basic arithmetic operations such as addition, multiplication, and division are used to compute the final value of the summation.
By applying these concepts, the problem-solving process involves extracting the constant, using the summation formula, substituting the values, and simplifying the expression step by step until the final result is obtained.
