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how fast is the the blockchain growing in MB/day or GB/month etc and what is the current blockchain size?

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There is no simple answer to this as it will vary wildly, but to give a ballpark figure, currently transactions average around 5.6kb (I got this by averaging the last 50 blocks from monerblocks.info), and with the current rate of transactions taken from moneroblocks.info we can see it's ~2 transactions per block. Given that the block transmission is every 2 mins, that means it grows ~4.03 megabytes per day simply from transactions based on current numbers. The current blockchain size is around 7.5Gb.

Papa Lazarou made a more accurate representation of monthly block growth in megabytes.

     Jan      Feb     Mar    Apr      May     Jun     Jul     Aug     Sep     Oct     Nov     Dec        
2014                          15.25   159.6   424.98  202.63  227.67  155.52  102.11  83.77    103.33   
2015  74.76   71.97   100.49  69.54   65.58   65.03   65.96   67.09   61.86   67.82   55.47      65     
2016  77.37   69.88   131.46  107.59  78.63   87.72   84.67   136.9   170.21  143.24 
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    7.5 GB is the size of the LMDB version of the blockchain, raw data and actual size is ~3.5 GB right now (downloads.getmonero.org/blockchain.raw). – user20 Oct 23 '16 at 20:53
  • The answer is incomplete, monthly growth is missing. Apart from that, I don't think the 50 blocks are representative, as this just covers the last 1h40m. – dpzz Oct 28 '16 at 17:34
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    Monthly growth added, and my sampling was not far off the monthly average as it adds up to about 120mb which is good enough as I explicitly stated it was a ballpark figure. The median of the last 2.5 years is 83mb and the last 6 months is 97mb. If we really want to split hairs, then I'd argue that this question is too broad and should be closed as it's literally impossible to give a concrete answer. All we can do is work on averages, medians and ultimately, guesswork. – ferretinjapan Oct 28 '16 at 18:08
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This is the blockchain growth (MB) per month since inception:

+--------+---------+
| size   | month   |
+--------+---------+
|  15.25 | 2014-04 |
| 159.60 | 2014-05 |
| 424.98 | 2014-06 |
| 202.63 | 2014-07 |
| 227.67 | 2014-08 |
| 155.52 | 2014-09 |
| 102.11 | 2014-10 |
|  83.77 | 2014-11 |
| 103.33 | 2014-12 |
|  74.76 | 2015-01 |
|  71.97 | 2015-02 |
| 100.49 | 2015-03 |
|  69.54 | 2015-04 |
|  65.58 | 2015-05 |
|  65.03 | 2015-06 |
|  65.96 | 2015-07 |
|  67.09 | 2015-08 |
|  61.86 | 2015-09 |
|  67.82 | 2015-10 |
|  55.47 | 2015-11 |
|  65.00 | 2015-12 |
|  77.37 | 2016-01 |
|  69.88 | 2016-02 |
| 131.46 | 2016-03 |
| 107.59 | 2016-04 |
|  78.63 | 2016-05 |
|  87.72 | 2016-06 |
|  84.67 | 2016-07 |
| 136.90 | 2016-08 |
| 170.21 | 2016-09 |
| 143.24 | 2016-10 |
+--------+---------+
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