Evaluate the Summation sum from j=1 to 6 of 10j
The question asks for the evaluation of a mathematical summation where the variable 'j' has values starting from 1 up to 6. For each value of 'j', it's multiplied by 10, and the resulting products are then added together to give the final sum. Essentially, you are requested to calculate the total sum of the sequence that arises when multiplying each number from 1 to 6 by 10, and then adding all those products together.
$\sum_{j = 1}^{6} 10 j$
Answer
Solution:
Step 1:
Extract the constant $10$ from the summation: $10 \sum_{j = 1}^{6} j$
Step 2:
Apply the arithmetic series sum formula: $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
Step 3:
Insert the upper limit of the summation into the formula, and don't forget to include the extracted constant: $10 \left( \frac{6(6 + 1)}{2} \right)$
Step 4:
Proceed with the simplification:
Step 4.1:
Begin by simplifying the inner expression:
Step 4.1.1:
Combine $6$ and $1$: $10 \left( \frac{6 \cdot 7}{2} \right)$
Step 4.1.2:
Calculate $6 \times 7$: $10 \left( \frac{42}{2} \right)$
Step 4.2:
Reduce the fraction by eliminating common factors:
Step 4.2.1:
Separate the $2$ from the $10$: $2 \cdot 5 \left( \frac{42}{2} \right)$
Step 4.2.2:
Eliminate the common factor of $2$: $5 \cdot 42$
Step 4.3:
Finish by multiplying $5$ by $42$: $210$
Knowledge Notes:
The problem involves evaluating a finite arithmetic series, which is a sequence of numbers with a constant difference between consecutive terms. The task is to find the sum of the first six terms of the series where each term is multiplied by $10$.
The arithmetic series sum formula for the first $n$ natural numbers is given by:
$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$ This formula is derived from the observation that the sum of the series is equivalent to the sum of the first and last term, the second and second-to-last term, and so on, each of which sums to $n + 1$. There are $\frac{n}{2}$ such pairs if $n$ is even (or $\frac{n+1}{2}$ pairs if $n$ is odd), hence the multiplication by $\frac{n}{2}$ or $\frac{n+1}{2}$.
In the given problem, the constant $10$ is factored out of the summation because of the distributive property of multiplication over addition, which states that $a \sum b = \sum ab$. This simplifies the calculation as it allows us to use the arithmetic series sum formula directly and then multiply the result by the constant factor.
After applying the formula, the problem becomes a matter of simplifying the resulting algebraic expression. This involves basic arithmetic operations: addition, multiplication, and division. The final step is to multiply the simplified factors to obtain the sum of the series.
