Which expression represents the sum of the finite series 13+10+7+4 ? I. ∑n=1^4(16-3 n) II. ∑n=3^

Which expression represents the sum of the finite series...

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence, which is characterized by a constant difference between consecutive terms. Understanding arithmetic series involves knowing how to determine the first term, the common difference, and how these values can be used to express each term of the series in a formula.

Summation Notation

Summation notation, represented by the sigma symbol (?), is a concise way of writing the sum of a sequence of terms. It allows for general terms to be summed over a given range of indices, making it easier to represent and manipulate finite series in mathematical expressions.

General Term of an Arithmetic Sequence

The general term of an arithmetic sequence is typically expressed using the formula a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. This concept is crucial for forming the series representation in summation notation by defining the term inside the summation.

Index Shifting in Summations

Index shifting in summations refers to changing the starting and ending indices of a summation expression without altering the actual sum. This concept is important when comparing different forms of summation expressions to ensure they represent the same series by properly aligning the indices and the corresponding terms.

Transcript

00:02 To find out which expression or expressions represent the finite sum of a series here, this 13 plus 10 plus 7 plus 4, we have to evaluate each of these summations down here to find out which of them generates the terms 13 plus 10 plus 7 plus 4.

00:23 So because it could be one of these or all of these, we have to check each and every one to find our answer.

00:30 Let's start off with this first one, option i.

00:34 Here we have the sum as n starts at 1 and goes to 4 for the expression 16 minus 3n.

00:42 So let's go ahead and start with n equals 1.

00:48 And if we have 16 minus 3 times 1, then we have 16 minus 3, which we know to be 13.

01:04 So so far we have the first term correct.

01:07 Let's go to the next one when n is equal to two.

01:12 Using our same expression, 16 minus three times two.

01:20 Three times two is six.

01:21 So we have 16 minus six, which is 10.

01:25 That's our second term.

01:27 So far so good.

01:29 When n is equal to three.

01:32 16 minus 3 times 3 3 times 3 we know is 9 16 minus 9 gives us 7 looks like we're on a roll one last one to check n equals 4 16 minus 3 times 4 3 times 4 gives us 12 so we have 16 minus 12 gives us our final term four.

02:08 So this summation would be 13 plus 10 plus 7 plus 4, which is indeed what our expression is.

02:17 So let's go ahead and give that one a check.

02:19 We know whichever option that we choose, it has to include option i.

02:25 Let's check our second expression.

02:28 Here we have the summation from when n starts at 3 and goes until it equals 6, with the expression is 22 minus 3n.

02:38 So if we start off with our n equals 3, and we have the expression 22 minus 3 times 3.

02:52 3 times 3 gives us 9.

02:53 So we have 22 minus 9, which will give us 13.

03:01 So far, so good.

03:04 Next, when n is equal to 4, we have 22.

03:10 Minus three times four.

03:16 Three times four we know is 12.