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# In an arithmetic sequence, the sum of the second and eighth terms is $5$, and the product of the fourth and fifth terms is also $5$. What i

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In an arithmetic sequence, the sum of the second and eighth terms is 5, and the product of the fourth and fifth terms is also 5 .

What is the 20th term of this sequence?

May 4, 2022

#1
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try writing everything in terms of a variable

May 4, 2022
#2
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Let a be the first term and d be the common difference of the arithmetic sequence.

$$\begin{cases} (a + d) + (a + 7d) = 5\\ (a + 3d)(a + 4d) = 5 \end{cases}$$

$$\begin{cases} a = \dfrac{5 - 8d}{2}\\ (a + 3d)(a + 4d) = 5 \end{cases}$$

Now you can substitute the first equation into the second, and solve the resulting quadratic equation for the value(s) of d.

Then you will also be able to find the corresponding value(s) of a.

The answer is a + 19d, but please calculate the value(s) of a and d and substitute to get the numerical answer.

May 4, 2022
#3
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Still can't get it. can you futhure elaborate

Guest May 4, 2022
#4
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Since a = (5 - 8d)/2, substituting this into the second equation will completely eliminate all "a"s in the second equation, giving $$\left(\dfrac{5 - 8d}2 + 3d\right)\left(\dfrac{5 - 8d}2+4d\right) = 5$$

Expand and simplify, and you will get an equation entirely in terms of d. Now you can solve it to get the value of d and a.

Once you get the value of d from that equation above, the value of a is just (5 - 8d)/2.

MaxWong  May 4, 2022
edited by MaxWong  May 4, 2022
#5
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Thank you. I realize that but I can't figure out d?

Guest May 4, 2022
#6
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What did you get after simplifying the equation?

MaxWong  May 4, 2022
#7
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i got 1/2 wat do i do next?

Guest May 7, 2022
#9
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You have d = 1/2, and the first equation just says a = (5 - 8d)/2. So you can substitute your answer into the first equation to get a.

MaxWong  May 8, 2022
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a_2+a_8=5, so that forces a_5 to be 5/2. a_4*5/2=5, so a_4 equals 2

May 8, 2022