In the following exercises, use summation properties and formulas to rewrite and evaluate the su

In the following exercises, use summation properties and...

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Key Concepts

Linearity of Summation

This concept refers to the ability to separate a complex summation into simpler parts by using properties like the sum of a difference or the sum of a scalar multiple. It allows one to rewrite a sum such as ?(a_j - b_j) into ?a_j ? ?b_j, and to pull constants out of the summation, simplifying the evaluation process.

Sum of Natural Numbers Formula

This formula, given by n(n + 1)/2 for the sum of the first n natural numbers, is a fundamental result in arithmetic series. It provides a quick way to evaluate sums of the form ?j, without having to add each individual term.

Sum of Squares Formula

This formula calculates the sum of the squares of the first n natural numbers, expressed as n(n + 1)(2n + 1)/6. It is essential for evaluating sums involving quadratic terms in the index, enabling a more efficient computation than manual summation.

Transcript

00:01 We're going to compute the value of this sum, number nine, and we're going to use these formulas off to the right, the red formula and the blue formula to help us, and some other formulas that i don't have visible right now, but they're the simple ones.

00:21 And they say we can take the sum of individual parts of a polynomial and find their results individually.

00:33 So i'm going to do that.

00:35 And then we can combine the results afterwards.

00:39 So from j equals 1 to 50 of just the first term of the polynomial.

00:52 Okay.

00:53 Minus just go ahead and switch to green here because they're actually blue because we're going to use the second formula for this part so j equals 1 250 of j now this 2 j now this 2j is the constant multiplier is allowed to be written out in front of the sum we can just sum up j and then multiply the result by 2.

01:33 So i've split this apart into the two formulas we're going to use.

01:39 And we can see here, switch back to red, that n is 50.

01:49 That's the top number of the sum.

01:52 So when i substitute this formula in over here for the power of 2, i'm going to put 50 in for n for n.

02:03 50.

02:05 And then right on down the next quantity is n plus 1.

02:09 So that's 51.

02:12 And the next quantity is 2n plus 1.

02:16 So that's 2 times 50 is 100 plus 1...