text: Cristofer Watson
  
  2022-10-25
  
  Find the number of terms in the following geometric series: 1 + 2 + 4 + ... + 67108864
  
  Do you have a similar question?
  
  Phillip Fletcher
  
  Expert
  
  The general form of a term of a geometric sequence or series is:
  ${a}_{n}=a{r}^{n-1}$
  where a is the first term and r is the common ratio.
  In our example, a=1 and r=2, so the question boils down to identifying which power of 2 is 67108864.
  Notice that ${2}^{10}=1024\approx 1000={10}^{3}$ and 67108864 is a little over $64\cdot {10}^{6}$ hence we find the correct power is:
  ${2}^{26}={2}^{6}\cdot {2}^{10}\cdot {2}^{10}=64\cdot 1024\cdot 1024$
  So there are 27 terms: ${2}^{0},{2}^{1},{2}^{2},...,{2}^{26}$
  
  Still Have Questions?
  
  Cory Russell
  
  Expert
  
  As a child, I used to like to write powers of 2 on a blackboard, starting with 1 and doubling it repeatedly.
  In later life I found it useful to memorise powers of 2 up to about ${2}^{32}=4294967296$
  A couple of 'fun' ones are ${2}^{25}=33554432$ and ${2}^{29}=536870912$ (which contains all the digits except 4).
  
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explanation: The document contains a math question and a detailed answer. The question asks for the number of terms in a geometric series, and the answer provides a step-by-step explanation of how to find the number of terms.
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distilabel_metadata.role: system
distilabel_metadata.content: Cristofer Watson
  
  2022-10-25
  
  Find the number of terms in the following geometric series: 1 + 2 + 4 + ... + 67108864
  
  Do you have a similar question?
  
  Phillip Fletcher
  
  Expert
  
  The general form of a term of a geometric sequence or series is:
  ${a}_{n}=a{r}^{n-1}$
  where a is the first term and r is the common ratio.
  In our example, a=1 and r=2, so the question boils down to identifying which power of 2 is 67108864.
  Notice that ${2}^{10}=1024\approx 1000={10}^{3}$ and 67108864 is a little over $64\cdot {10}^{6}$ hence we find the correct power is:
  ${2}^{26}={2}^{6}\cdot {2}^{10}\cdot {2}^{10}=64\cdot 1024\cdot 1024$
  So there are 27 terms: ${2}^{0},{2}^{1},{2}^{2},...,{2}^{26}$
  
  Still Have Questions?
  
  Cory Russell
  
  Expert
  
  As a child, I used to like to write powers of 2 on a blackboard, starting with 1 and doubling it repeatedly.
  In later life I found it useful to memorise powers of 2 up to about ${2}^{32}=4294967296$
  A couple of 'fun' ones are ${2}^{25}=33554432$ and ${2}^{29}=536870912$ (which contains all the digits except 4).
  
  Free Math Solver
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer. The question asks for the number of terms in a geometric series, and the answer provides a step-by-step explanation of how to find the number of terms.
  
  Label: question-and-answer