Evaluate the Summation sum from i=1 to 21 of (i-8)^2
The problem provided is a mathematical summation problem that requires calculating the sum of the squares of a series of numbers, with a specific adjustment to each term in the series. The question asks the respondent to evaluate the cumulative total of the given formula: (i - 8)^2, where i represents each integer starting from 1 and ending with 21. For each value of i within this range, the formula dictates to subtract 8 from i, square the result, and then add it to the sum total. The task is to perform this operation for each integer value of i from 1 to 21 and calculate the final sum.
$\sum_{i = 1}^{21} \left(\left(\right. i - 8 \left.\right)\right)^{2}$
Answer
Solution:
Step 1: Simplify the given summation expression.
Step 1.1: Express $(i - 8)^2$ as a product of two identical binomials: $(i - 8)(i - 8)$.
Step 1.2: Expand the binomial using the distributive property (FOIL method).
Step 1.2.1: Distribute $i$ across the binomial: $i(i - 8) - 8(i - 8)$.
Step 1.2.2: Continue distribution: $i \cdot i - i \cdot 8 - 8 \cdot i + 8 \cdot 8$.
Step 1.2.3: Complete the distribution: $i^2 - 8i - 8i + 64$.
Step 1.3: Combine like terms in the expanded expression.
Step 1.3.1: Simplify individual terms.
Step 1.3.1.1: $i \cdot i$ simplifies to $i^2$.
Step 1.3.1.2: $-8i$ and $-8i$ combine to $-16i$.
Step 1.3.1.3: $8 \cdot -8$ simplifies to $+64$.
Step 1.3.2: Combine like terms: $i^2 - 16i + 64$.
Step 1.4: Rewrite the summation with the simplified expression: $\sum_{i=1}^{21} (i^2 - 16i + 64)$.
Step 2: Break down the summation into individual summations: $\sum_{i=1}^{21} i^2 - 16\sum_{i=1}^{21} i + \sum_{i=1}^{21} 64$.
Step 3: Evaluate the summation of squares $\sum_{i=1}^{21} i^2$.
Step 3.1: Use the formula for the sum of squares: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.
Step 3.2: Insert the upper limit of the summation into the formula: $\frac{21(21+1)(2 \cdot 21+1)}{6}$.
Step 3.3: Simplify the expression.
Step 3.3.1: Reduce the fraction by canceling common factors.
Step 3.3.1.1: Extract a factor of 3: $\frac{3 \cdot 7(21+1)(2 \cdot 21+1)}{6}$.
Step 3.3.1.2: Cancel the common factor of 3: $\frac{7(21+1)(2 \cdot 21+1)}{2}$.
Step 3.3.2: Perform arithmetic operations in the numerator.
Step 3.3.2.1: Multiply 2 by 21.
Step 3.3.2.2: Add 21 and 1.
Step 3.3.2.3: Multiply 7 by 22.
Step 3.3.2.4: Add 42 and 1.
Step 3.3.3: Finalize the simplification.
Step 3.3.3.1: Multiply 154 by 43.
Step 3.3.3.2: Divide 6622 by 2 to get 3311.
Step 4: Evaluate the summation of the first 21 integers $16\sum_{i=1}^{21} i$.
Step 4.1: Use the formula for the sum of the first $n$ integers: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$.
Step 4.2: Insert the upper limit of the summation into the formula and multiply by the coefficient: $-16\left(\frac{21(21+1)}{2}\right)$.
Step 4.3: Simplify the expression.
Step 4.3.1: Perform arithmetic operations.
Step 4.3.1.1: Add 21 and 1.
Step 4.3.1.2: Multiply 21 by 22.
Step 4.3.2: Reduce the fraction by canceling common factors.
Step 4.3.2.1: Factor out 2 from -16.
Step 4.3.2.2: Cancel the common factor of 2.
Step 4.3.2.3: Rewrite the expression as $-8 \cdot 462$.
Step 4.3.3: Multiply -8 by 462 to get -3696.
Step 5: Evaluate the summation of the constant 64 over 21 terms $\sum_{i=1}^{21} 64$.
Step 5.1: Use the formula for the sum of a constant: $\sum_{k=1}^{n} c = cn$.
Step 5.2: Insert the values into the formula: $64 \cdot 21$.
Step 5.3: Multiply 64 by 21 to get 1344.
Step 6: Combine the results of the individual summations: $3311 - 3696 + 1344$.
Step 7: Simplify the final result.
Step 7.1: Subtract 3696 from 3311 to get -385.
Step 7.2: Add -385 to 1344 to get the final answer of 959.
Knowledge Notes:
Summation Notation: Summation notation is a way to represent the sum of a sequence of terms, denoted by the Greek letter sigma ($\Sigma$). It includes an expression for the terms to be summed, an index of summation with a starting value, and an upper limit.
Binomial Expansion: The square of a binomial $(a - b)^2$ can be expanded using the distributive property (FOIL method) to $a^2 - 2ab + b^2$.
Summation Formulas: There are formulas for the sum of the first $n$ natural numbers, the sum of the squares of the first $n$ natural numbers, and the sum of a constant sequence, which are:
- Sum of the first $n$ natural numbers: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
- Sum of the squares of the first $n$ natural numbers: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$
- Sum of a constant sequence: $\sum_{k=1}^{n} c = cn$
Combining Like Terms: In algebra, combining like terms is a process used to simplify an expression or to add or subtract polynomials. Like terms are terms that have the same variables raised to the same power.
Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are used to simplify expressions and to calculate the value of summations.
