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 a10 + b10.

Accepted Answer

well everyone in this video, we're going to be looking at this practice problem where we're given two unique sequences, sequence A which has negative 36, negative 12 and positive 24. And sequence B which has four negative six, negative 16 and -26. And we want to find the eighth term of A. And add it with the eighth term of B. So in this case in order to find the eighth term or in order to add the eighth term of both the A and B. Sequence we first want to determine what sort of sequences we have. So the first sequence A you can see that we have negative three and then positive six and then a negative again and then a positive. So because it alternates between positive and negative, this gives us a clue that we're multiplying this the previous numbers with a negative value and then again with the same negative value. So this is a geometric sequence. And we can figure out this ratio that we're multiplying each previous term by by dividing the second term six by the first term negative three. So I'm gonna say the ratio is r and r is equal to six divided by negative three. Which means that R is equal to -2. So if I look back at the sequence negative three times negative two is positive six and positive six times negative two is negative 12. So this ratio of negative two makes sense. And if we look at this sequence B, you can see that once we go negative we don't go back to positive. And the difference between 16 and 26 is 10 and six and 16 is 10. So I have a suspicion that this is going to be an earth metric sequence. And with their metric sequences we have a difference That I'll represent as D. And we're going to find it similar to geometric sequence where we have the second term in this case -6. But now I'm gonna subtract the first term four. So this is telling me that there's a difference of negative 10 between each term. And if I look back at my sequence I have four minus 10 is negative six and negative six minus 10 is negative 16 and so on. So that makes sense. So now I figured out that I have a geometric and arithmetic sequence. I figured out the ratio for the geometric sequence and the difference for the arithmetic sequence. So now we can apply the formulas that we know for each of these. So for a geometric sequence we have the formula that A of n is equal to A one times are times Our Times or to the Power of N -1. Where N is equal to the term we want In this case we want the 8th term, A one is the first term which is negative three and r is the ratio which we've already determined to be negative too. So we can go ahead and plug those values in to find the eighth term of the sequence A. So the eighth term is equal to The first term negative three times are negative too to the end or 8 -1 power. If I simplify this I have negative three times negative two to the seventh power And -2 to the 7th power is -128. So that means that the eighth term of the sequence A is negative three times negative 1 28 which is negative or positive 384. So we're going to follow the same pattern now for the B sequence which is an arithmetic sequence. And when we're trying to find a certain term for that, the formula is A. N. Is equal to a one plus n minus one times D. And again and is equal to the term that we want. Which again the 8th term A one is equal to the first term which is in this case four and D. Is the difference that we already solved or negative 10. So I can go ahead and plug those into my formula. So the I'm going to re label A as B. So I don't get confused because I'm looking at the B sequence. So B eight is equal to the first term four plus N which is eight minus one times D. Which is negative 10. If I simplify this I still have four plus seven times negative 10 for minus 70 negative 66. So be or the eighth term of the sequence B is negative 66. And the eighth term of the A sequence is 3 84. And I can go ahead and go back to my original equation that I'm trying to solve, which is eighth term of a plus eighth term of B. So I have 3 84 plus negative 66. And if I do this math, I got 318. So this becomes my final answer for this problem. And you can see that this aligns with answer choice A. So hopefully this video hope 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 a10 + b10.

Key Concepts

Sequences

Arithmetic and Geometric Sequences

Summation of Terms

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