In the sequence 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8 where n consective terms have the value n, then 1025th term is?

1 Answer
KillerBunny
Jun 24, 2018

#1024#

Explanation:

This sequence is composed by powers of #2#, repeated #2^n# times.

In fact, you have #2^0=1#, and #1# appears one time.

Then, #2^1=2#, and #2# appears twice.

Then, #2^2=4# and #4# appears four times.

Then, #2^3=8#, and #8# appears eight times.

So, the next number will be the next power of #2#, and it will appear #2^n# times.

Also, note that the first #1# is in first position, the first #2# is in second position, the first #4# is in fourth position, and so on. This makes it easy to write the sequence without repetition, like this:

  • There is one #1#, starting from position #1#
  • There are two #2#s, starting from position #2#
  • There are four #4#s, starting from position #4#
  • There are eight #8#s, starting from position #8#
  • ...
  • There are five hundred and twelve #512#s, starting from position #512#
  • There are one thousand and twenty-four #1024#s, starting from position #1024#

So, the #1025^{"th"}# element is #1024#.

In general, the #k^{th}# term of this sequence is #2^n#, if #2^n \le k < 2^{n+1}#

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