Probability - Geometric Sequence ProblemI have been trying to solve a problem which states the following:For every hour less than 8, an initial value depreciates by 7.9%. The initial value given is 87. This is what I have calculated so far:8 hours = 877 hours = 80.1276 hours = 73.7975 hours = 67.9674 hours = 62.5983 hours = 57.6522 hours = 53.0981 hours = 48.903From the above, it can be seen that the problem is a geometric sequence, having a constant ratio of 0.921.The problem asks to find the number of hours it would take to reach 50. I know that multiplying a previous value by 0.921 gives you the next value for a whole hour (bottom to top). However, I am really struggling to find a way to calculate the hours for the value 50. How can I go about doing this?

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yatangije62 · Accepted Answer

Step 1After each hour, the value V is multiplied by 0.921 (i.e. losing 7.9%).So after h hours, the value is the initial value, 87, multiplied by 0.921, h times:  V  =  87  ∗  0.921  ∗  0.921  ∗  .  .  .  ∗  0.921 (h occurences of 0.921)This can be written as   V  =  87  ∗      0.921    h  , and the formula works for fractional values of h as well as for whole numbers.Step 2Rearrange this formula by taking logarithms of both sides (noting that   l  o  g  (  A  ∗  B  )  =  l  o  g  (  A  )  +  l  o  g  (  B  ) and   l  o  g  (      A    C    )  =  C  ∗  l  o  g  (  A  )So:   l  o  g  (  V  )  =  l  o  g  (  87  )  +  h  ∗  l  o  g  (  0.921  )You can then substitute in the value   V  =  50 to solve for h

Find the number of hours it would take to reach 50. I know that multiplying a previous value by 0.921 gives you the next value for a whole hour (bottom to top). However, I am really struggling to find a way to calculate the hours for the value 50. How can I go about doing this?

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