2,2,4 12,16,80, ..... what comes after 80 and whay ?

Question

2,2,4 12,16,80, What comes after 80 and how pls

Answer 1

If we call the terms a₀, a₁, a₂, a₃, a₄, a₅ and group them in pairs: (a₀, a₁), (a₂, a₃), (a₄, a₅) we can relate the second member of each pair to its partner: a[2k+1]=a[2k](2k+1) where the square brackets represent the subscript and k starts at 0. So, for example, if k=1, a₃=3a₂ and if k=2, a₅=5a₄.

Also a[2k]=2^(2^k), so, for example, if k=0, a₀=2^(2⁰)=2¹=2 and if k=1, a₂=2^(2¹)=2²=4, and a₁=a₀×1=2; if k=1, a₃=3a₂=3×4=12. So we have the series: 2, 2, 4, 12, 16, 80, the given series. 

Therefore, a₆ is found from k=3: 2^(2³)=2⁸=256. The series becomes 2, 2, 4, 12, 16, 80, 256. (The next term would be 7×256=1792.)

This logic can be simplfied:

First, write down the first, third and fifth term but leave a gap between them:

2

4

16

(2 multiplied by itself is 4, 4 multiplied by itself is 16.)

Next, write the numbers 1 to 6 underneath, including the gaps:

2		4		16
1	2	3	4	5	6

To fill in the gaps, multiply the number on the top line by the one beneath it and put the result where the gap next to it is:

2	2	4	12	16	80
×	=↑	×	=↑	×	=↑
1	2	3	4	5	6

So, 2×1=2 goes into the first gap; 4×3=12 goes into the next gap; 16×5=80. That gives us the series. So to continue, we would, by this logic, need 16 multiplied by itself=256.

Comments

What logic was used to get 86?

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Then put your teacher on the spot and ask him/her to explain it to you. After all, they asked you to do the question! Then present the solution I came up with (if they let you) and see what they say. I have at least provided some logic in support of the solution, even though it’s not what the book or automated solution says. If you are in Cyberschool, you can use their feedback system to explain the different solution with supporting logic.

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Maybe some other user on Math Homework Answers will come up with 86 as the solution with the accompanying logic. It’s probably quite simple, but I missed it and came up with something different! Thanks for replying to my earlier comments.

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I wish you success in your exam. Thank you for telling me that you’re studying for college entrance. If I can help you at any time you can also send me a private message, although I can’t guarantee I could answer all your questions! I would love to hear how you get on, particularly if you get into college.