Evaluate the Summation sum from n=1 to 7 of 5n
The problem involves calculating the total sum of a sequence of numbers that follow a specific rule. In this sequence, every term is generated by multiplying the position of the term in the sequence (denoted as 'n') by the number 5. The operation must be performed for every value of 'n' starting from 1 and ending at 7 (inclusive). To solve this, you would need to apply the summation process to each term, which, in this case, can be expressed mathematically as 5 times 1, 5 times 2, and so on, up to 5 times 7. The question asks for the sum of these products.
$\sum_{n = 1}^{7} 5 n$
Answer
Solution:
Step 1:
Extract the constant $5$ from the summation: $5 \sum_{n = 1}^{7} n$
Step 2:
Utilize the summation formula for a first-degree polynomial: $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
Step 3:
Insert the upper limit of the summation into the formula, and don't forget to include the constant we factored out: $5 \left( \frac{7(7 + 1)}{2} \right)$
Step 4:
Proceed with the simplification:
Step 4.1:
Combine $7$ and $1$: $5 \times \frac{7 \times 8}{2}$
Step 4.2:
Calculate $7$ times $8$: $5 \left( \frac{56}{2} \right)$
Step 4.3:
Divide $56$ by $2$: $5 \times 28$
Step 4.4:
Multiply $5$ by $28$: $140$
Knowledge Notes:
The problem involves evaluating a finite arithmetic series, which is a sequence of numbers with a constant difference between consecutive terms. In this case, the series is $5, 10, 15, ..., 35$, which can be represented as $5n$ for $n=1$ to $n=7$.
The process of solving this problem includes:
Factoring out constants from the summation: When a constant is multiplied by each term in a series, it can be factored out to simplify the summation process.
Using the summation formula for arithmetic series: The formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$ is a well-known result for the sum of the first $n$ natural numbers. This formula is derived from the observation that the sum of the series is equivalent to the sum of $n$ pairs of numbers each equal to $n+1$.
Substitution: After factoring out constants and identifying the correct summation formula, the next step is to substitute the upper limit of the summation into the formula.
Simplification: The final step involves basic arithmetic operations鈥攁ddition, multiplication, and division鈥攖o arrive at the final result.
Understanding these steps and the underlying principles of arithmetic series is essential for solving similar problems in mathematics.
