In Exercises 57–62, let
{a_n} = - 5, 10, - 20, 40, ...,
{b_n} = ...

Question

In Exercises 57–62, let
{a_n} = - 5, 10, - 20, 40, ..., 
{b_n} = 10, - 5, - 20, - 35, ...,
{c_n} = - 2, 1, - 1/2, 1/4

Find the difference between the sum of the first 10 terms of {an} and the sum of the first 10 terms of {bn}.

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we have two sequences. The first one A is equal to negative 36 negative 24. And the second one B is equal to four. Negative six. Negative 16 negative 26. And we're trying to find the difference between The sum of the 1st 12 terms of a. and B sequence. So essentially we want to find for the first sequence A. The sum Up to the 12th term and subtract that from And subtract the sum of the 1st 12 terms from the B sequence. So I've sort of denoted an expression here where I have the sum of The 1st 12 terms of the a sequence minus the sum of the 1st 12 terms of the B sequence. So essentially this is what I'm trying to find. But in order to do that I need to find the sum of each of the sequences independently. First. And if I look at the A sequence you can see that I alternate between negative and then positive numbers. So that to me is an indication that I'm going to be multiplying by a ratio that has a negative sign, which means that A is most likely a geometric sequence. And I can find this ratio by dividing the second term by the first term. So if I do that my ratio is equal to six divided by negative three. The second term divided by the first term. And from that I get that the ratio is negative two. If I look at my second sequence B, you can see that I'm no longer alternating between positive or negative values and actually between 16 and 26, it seems to only decrease by 10. And from negative six to negative 16 it decreases by 10 again. So it seems like I'm just subtracting 10. So it seems like arithmetic sequence, which means there's a difference between my terms that I'm just subtracting or adding and I can find that by doing the difference between my second term negative six minus my first term which is four. And that gives me negative 10. As I observed before, it seems to be subtracting 10 as the terms increased. So from negative six to negative 16, I subtracted 10 from negative six And so so on for the -26 as well. So now I've established that sequence A is geometric and sequence B is arithmetic and I found the ratio and the difference for each of those. So I can use the formulas that they have for the some. So when I have an geometric sequence I can find the sum of that sequence with the formula S. F N is equal to A one times are to the end power -1 divided by r minus one. And in this case And is equal to the term we're trying to add up to which we're trying to do the 1st 12 terms. So N is equal to 12. R. Is the ratio which we already sold for to be negative too. And a one is the first term Which is -3. So knowing all these values, I can plug them into the equation. So I get the sum of the 1st 12 terms is equal to the first term negative three times negative two to the end power. So 12 -1, Divided by R -1. So -2 -1. And if I simplify that I get As to the 12 is equal to negative three times negative two is negative, two should be in parentheses negative two to the 12th. Power is minus one, Divided by -2 -1 is -3. And here you can see that these negative threes cancel out to be one. So I'm left with The sum of the 1st 12 terms is equal to 4,096 -1 Which is 4095. So this is for my a sequence. And now we need to figure out the sum for my second sequence. And we can also use formulas established for the arithmetic took sequence. And that formula follows the form S of N is equal to n divided by two times two times be one Plus and -1 times d. and again N is equal to the term I'm trying to go up to which is 12 and B one is equal to the first term of the sequence which is four and D is the difference which we already calculated to be negative 10. So again I can plug in the values, so some of the 1st 12 terms becomes and divided by two. So 12 divided by two Times two times B one which is four Plus and -1. So 12 -1 times d. Which is negative 10. And if I simplify this 12 divided by two is times four is eight plus 12 minus one is 11 and then 11 times negative, 10 is negative 110. So I have S 12 is equal to six times eight plus negative 1, 10 is negative 102. So the sum of the 1st 12 terms of the b sequences, -612. And that's for the B sequence and now I can go back to what we're trying to find which is the difference is the difference of the sum. So I have my a sequence is 95 minus the sum of the B sequence which is negative 6 12. So this becomes 4095. The negatives cancel out. So I have plus 612 And that is equal to 707. So this becomes my final answer for the Difference is the difference between sums of the 1st 12 terms of the a. And the b. Sequence. And you can see that this aligns with answer choice a. So hopefully this video helped and see you next time.

In Exercises 57–62, let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find the difference between the sum of the first 10 terms of {an} and the sum of the first 10 terms of {bn}.

Key Concepts

Arithmetic Sequences

Summation of Series

Difference of Sums

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