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the answer is 364!

Sorry, all of my addition symbols were stripped out of my previous post – don’t really know why. The patterns should read:

4(plus16=),20(plus36=),56(plus64=),120(plus100=),220

and

4(plus16=),20(plus36=),56(plus64=),120(plus100=),220(plus144=),364

OK, perhaps it’s a roundabout way to get there, but here it is…

Original sequence: 4,20,56,120,220,…

Identify the sequence pattern: 4(16=),20(36=),56(64=),120(100=),220,… I do realize these are not actual mathematical equations, I’m just trying to illustrate the pattern.

Each number that is added to the previous number to get the next is a perfect square.

When reduced to it’s square root, a simpler pattern is revealed: 4,6,8,10,…

The next number in this simple sequence is obviously 12.
12 squared is 144. Therefore:

4(16=),20(36=),56(64=),120(100=),220(+144=),364

364 is the next number in the sequence.

I desist

An+1 = An + 2^{(n+3) if n is odd
An+1 = An + (n-1)*2}5 + 4 if n is even
A0 = 4

Almost! Last attempt…

An+1 = An + 2(n+3) if n is odd
An+1 = An + (n-1)*2^5 + 4 if n is even
A0 = 4

I’m sorry, I didn’t know the formatting. Another attempt…

An+1 = An + 2*(n+3) if n is odd
An+1 = An + (n-1)*25 + 4 if n is even
A0 = 4

I reached a different solution, which yields to a different number.

Mine is:

An+1 = An + 2^{(n+3) if n is odd
An+1 = An + (n-1)*2}5 + 4 if n is even
A0 = 4

A1 = 4 + 2^{4 = 20
A2 = 20 + 1*2}5 + 4 = 56
A3 = 56 + 2^{6 = 120
A4 = 120 + 3*2}5 + 4 = 220
A5 = 220 + 2^8 = 476

easy way to find is to find out diff between number
1st diff set – 16(20-4), 36(56-20), 64(100-56), 100(220-120)
2nd diff- 20(36-16), 28(64-36), 36(100-64),now answer is easy one it must be 44 to continue in same 8 number diff.. go reverse way 44 now answer is (44+100+220) = 364

Where do you post the answer?

Never have I ever gotten a difficult question like this right and I don’t understand how anyone can

Is there a way to arrive at the solution without trial and error?

364